We show that every internal biequivalence in a tricategory T is part of a biadjoint biequivalence. We give two applications of this result, one for transporting monoidal structures and one …
We show that every internal biequivalence in a tricategory T is part of a biadjoint biequivalence. We give two applications of this result, one for transporting monoidal structures and one for equipping a monoidal bicategory with invertible objects with a coherent choice of those inverses.
We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called …
We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical bicategory, this being a bicategory enriched in the monoidal 2-category of pseudo double categories. Finally, we show that every sufficiently well-behaved locally cubical bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.
Abstract A multivariable adjunction is the generalisation of the notion of a 2-variable adjunction, the classical example being the hom/tensor/cotensor trio of functors, to n + 1 functors of n …
Abstract A multivariable adjunction is the generalisation of the notion of a 2-variable adjunction, the classical example being the hom/tensor/cotensor trio of functors, to n + 1 functors of n variables. In the presence of multivariable adjunctions, natural transformations between certain composites built from multivariable functors have “dual” forms. We refer to corresponding natural transformations as multivariable or parametrised mates, generalising the mates correspondence for ordinary adjunctions, which enables one to pass between natural transformations involving left adjoints to those involving right adjoints. A central problem is how to express the naturality (or functoriality) of the parametrised mates, giving a precise characterization of the dualities so-encoded. We present the notion of “cyclic double multicategory” as a structure in which to organise multivariable adjunctions and mates. While the standard mates correspondence is described using an isomorphism of double categories, the multivariable version requires the framework of “double multicategories”. Moreover, we show that the analogous isomorphisms of double multicategories give a cyclic action on the multimaps, yielding the notion of “cyclic double multicategory”. The work is motivated by and applied to Riehl's approach to algebraic monoidal model categories.
Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structurepreserving maps, but often the maps of interest are merely lax monoidal.We describe conditions under which …
Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structurepreserving maps, but often the maps of interest are merely lax monoidal.We describe conditions under which one can transport the weak equivalences from one category to another with the same objects and a broader class of maps.Under mild hypotheses this process produces an equivalence of homotopy theories.We describe examples including algebras over an operad, such as symmetric monoidal categories and n-fold monoidal categories; and diagram categories, such as Γ-categories.
We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and …
We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and degenerate bicategories to monoidal categories; however, to understand this correspondence fully we examine the totalities of such structures together with maps between them and higher maps between those. Categories naturally form a 2-category {\bfseries Cat} so we take the full sub-2-category of this whose 0-cells are the degenerate categories. Monoids naturally form a category, but we regard this as a discrete 2-category to make the comparison. We show that this construction does not yield a biequivalence; to get an equivalence we ignore the natural transformations and consider only the {\it category} of degenerate categories. A similar situation occurs for degenerate bicategories. The tricategory of such does not yield an equivalence with monoidal categories; we must consider only the categories of such structures. For doubly degenerate bicategories the tricategory of such is not naturally triequivalent to the category of commutative monoids (regarded as a tricategory). However in this case considering just the categories does not give an equivalence either; to get an equivalence we consider the {\it bicategory} of doubly degenerate bicategories. We conclude with a hypothesis about how the above cases might generalise for n-fold degenerate n-categories.
Nous continuons le travail commence en [5] en etudiant les tricategories degenerees et en les comparant avec les structures predites par le tableau periodique des n-categories. Pour les tricategories trois …
Nous continuons le travail commence en [5] en etudiant les tricategories degenerees et en les comparant avec les structures predites par le tableau periodique des n-categories. Pour les tricategories trois fois degenerees nous demontrons une triequivalence avec la tricategorie partiellement discrete des monoides commutatifs. Pour les tricategories deux fois degenerees nous expliquons comment on peut construire une categorie monoidale tressee d'une tricategorie deux fois degeneree donnee, mais nous demontrons que cette construction n'induit pas une comparaison simple entre BrMonCat et Tricat. Nous discutons comment on peut iterer la construction des icones pour produire un equivalence, mais nous esperons a la suite pour donner les details. Finalement nous etudions les tricategories degenerees pour donner la premiere definition de bicategorie monoidale completement algebrique et la structure entiere de tricategorie de MonBicat. We continue the project begun in [5] by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. For triply degenerate tricategories we exhibit a triequivalence with the partially discrete tricategory of commutative monoids. For the doubly degenerate case we explain how to construct a braided monoidal category from a given doubly degenerate category, but show that this does not induce a straightforward comparison between BrMonCat and Tricat. We indicate how to iterate the icon construction to produce an equivalence, but leave the details to a sequel. Finally we study degenerate tricategories in order to give the first fully algebraic definition of monoidal bicategories and the full tricategory structure MonBicat.
We continue the project begun in ``The periodic table of $n$-categories for low dimensions I'' by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. …
We continue the project begun in ``The periodic table of $n$-categories for low dimensions I'' by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. For triply degenerate tricategories we exhibit a triequivalence with the partially discrete tricategory of commutative monoids. For the doubly degenerate case we explain how to construct a braided monoidal category from a given doubly degenerate category, but show that this does not induce a straightforward comparison between \bfseries{BrMonCat} and \bfseries{Tricat}. We show how to alter the natural structure of \bfseries{Tricat} in two different ways to provide a comparison, but show that only the more brutal alteration yields an equivalence. Finally we study degenerate tricategories in order to give the first fully algebraic definition of monoidal bicategories and the full tricategory structure \bfseries{MonBicat}.
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study …
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study the 2-category of algebras over an operad in Cat. We show that any operad is finitary, that an operad is cartesian if and only if the group actions are nearly free (in a precise fashion), and that the existence of a pseudo-commutative structure largely depends on the groups of equivariance. We conclude by showing that the operad for strict braided monoidal categories has two canonical pseudo-commutative structures.
We form tricategories and the homomorphisms between them into a bicategory. We then enrich this bicategory into an example of a three-dimensional structure called a locally double bicategory, this being …
We form tricategories and the homomorphisms between them into a bicategory. We then enrich this bicategory into an example of a three-dimensional structure called a locally double bicategory, this being a bicategory enriched in the monoidal 2-category of weak double categories. Finally, we show that every sufficiently well-behaved locally double bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.
Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1and 2-cells.The classifying space of a Picard 2-category D is an infinite loop space, the zeroth space of the K -theory …
Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1and 2-cells.The classifying space of a Picard 2-category D is an infinite loop space, the zeroth space of the K -theory spectrum KD.This spectrum has stable homotopy groups concentrated in levels 0, 1 and 2. We describe part of the Postnikov data of KD in terms of categorical structure.We use this to show that there is no strict skeletal Picard 2-category whose K -theory realizes the 2-truncation of the sphere spectrum.As part of the proof, we construct a categorical suspension, producing a Picard 2-category †C from a Picard 1-category C , and show that it commutes with K -theory, in that K †C is stably equivalent to †KC .
In the study of higher categories, dimension three occupies an interesting position on the landscape of higher dimensional category theory. From the perspective of a “hands-on” approach to defining weak …
In the study of higher categories, dimension three occupies an interesting position on the landscape of higher dimensional category theory. From the perspective of a “hands-on” approach to defining weak n-categories, tricategories represent the most complicated kind of higher category that the community at large seems comfortable working with. On the other hand, dimension three is the lowest dimension in which strict n-categories are genuinely more restrictive than fully weak ones, so tricategories should be a sort of jumping off point for understanding general higher dimensional phenomena. This work is intended to provide an accessible introduction to coherence problems in three-dimensional category.
We present the notion of "cyclic double multicategory", as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2-variable adjunction is the hom/tensor/cotensor trio …
We present the notion of "cyclic double multicategory", as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2-variable adjunction is the hom/tensor/cotensor trio of functors; we generalise this situation to n+1 functors of n variables. Furthermore, we generalise the mates correspondence, which enables us to pass between natural transformations involving left adjoints to those involving right adjoints. While the standard mates correspondence is described using an isomorphism of double categories, the multivariable version requires the framework of "double multicategories". Moreover, we show that the analogous isomorphisms of double multicategories give a cyclic action on the multimaps, yielding the notion of "cyclic double multicategory". The work is motivated by and applied to Riehl's approach to algebraic monoidal model categories.
We study the monoidal structure of the standard strictification functor $\textrm{st}:\mathbf{Bicat} \rightarrow \mathbf{2Cat}$. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the …
We study the monoidal structure of the standard strictification functor $\textrm{st}:\mathbf{Bicat} \rightarrow \mathbf{2Cat}$. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the 2-category whose objects are 2-categories.
We study the totality of weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric …
We study the totality of weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric then this process can be iterated. We show that starting from the symmetric monoidal bicategory Cat and performing the construction twice yields a convenient symmetric monoidal bicategory of partially strict tricategories. We show that restricting to the doubly degenerate ones immediately gives the correct bicategory of 2-tuply monoidal categories missing from our earlier studies of the Periodic Table. We propose a generalisation to all k-tuply monoidal n-categories.
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study …
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study the 2-category of algebras over an operad in Cat. We show that any operad is finitary, that an operad is cartesian if and only if the group actions are nearly free (in a precise fashion), and that the existence of a pseudo-commutative structure largely depends on the groups of equivariance. We conclude by showing that the operad for strict braided monoidal categories has two canonical pseudo-commutative structures.
It was argued by Crans that it is too much to ask that the category of Gray-categories admit a well behaved monoidal biclosed structure. We make this precise by establishing …
It was argued by Crans that it is too much to ask that the category of Gray-categories admit a well behaved monoidal biclosed structure. We make this precise by establishing undesirable properties that any such monoidal biclosed structure must have. In particular we show that there does not exist any tensor product making the model category of Gray-categories into a monoidal model category.
It was argued by Crans that it is too much to ask that the category of Gray-categories admit a well behaved monoidal biclosed structure. We make this precise by establishing …
It was argued by Crans that it is too much to ask that the category of Gray-categories admit a well behaved monoidal biclosed structure. We make this precise by establishing undesirable properties that any such monoidal biclosed structure must have. In particular we show that there does not exist any tensor product making the model category of Gray-categories into a monoidal model category.
We show that every action operad gives rise to a notion of monoidal category via the categorical version of the Borel construction, embedding action operads into the category of 2-monads …
We show that every action operad gives rise to a notion of monoidal category via the categorical version of the Borel construction, embedding action operads into the category of 2-monads on $\mathbf{Cat}$. We characterize those 2-monads in the image of this embedding, and as an example show that the theory of coboundary categories corresponds precisely to the operad of $n$-fruit cactus groups. We finally define $\mathbf{\Lambda}$-multicategories for an action operad $\mathbf{\Lambda}$, and show that they arise as monads in a Kleisli bicategory.
We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpretted topologically …
We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpretted topologically using up-to-homotopy operad actions and the algebraic classification of surface braids.
We show that every action operad gives rise to a notion of monoidal category via the categorical version of the Borel construction, embedding action operads into the category of 2-monads …
We show that every action operad gives rise to a notion of monoidal category via the categorical version of the Borel construction, embedding action operads into the category of 2-monads on $\mathbf{Cat}$. We characterize those 2-monads in the image of this embedding, and as an example show that the theory of coboundary categories corresponds precisely to the operad of $n$-fruit cactus groups. We finally define $\mathbf{\Lambda}$-multicategories for an action operad $\mathbf{\Lambda}$, and show that they arise as monads in a Kleisli bicategory.
We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and …
We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and degenerate bicategories to monoidal categories; however, to understand this correspondence fully we examine the totalities of such structures together with maps between them and higher maps between those. Categories naturally form a 2-category {\bfseries Cat} so we take the full sub-2-category of this whose 0-cells are the degenerate categories. Monoids naturally form a category, but we regard this as a discrete 2-category to make the comparison. We show that this construction does not yield a biequivalence; to get an equivalence we ignore the natural transformations and consider only the {\it category} of degenerate categories. A similar situation occurs for degenerate bicategories. The tricategory of such does not yield an equivalence with monoidal categories; we must consider only the categories of such structures. For doubly degenerate bicategories the tricategory of such is not naturally triequivalent to the category of commutative monoids (regarded as a tricategory). However in this case considering just the categories does not give an equivalence either; to get an equivalence we consider the {\it bicategory} of doubly degenerate bicategories. We conclude with a hypothesis about how the above cases might generalise for n-fold degenerate n-categories.
Constructions of spaces from symmetric monoidal categories or $\Gamma$-categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal or lax …
Constructions of spaces from symmetric monoidal categories or $\Gamma$-categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal or lax natural. We describe conditions under which one can transport the weak equivalences from such a category of structured objects and strict maps to the corresponding category with the same objects and lax maps. Under mild hypotheses this process produces an equivalence of homotopy theories. We describe examples including algebras over an operad, such as symmetric monoidal categories and $n$-fold monoidal categories; and diagram categories, such as $\Gamma$-categories.
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The study of codescent in Gray-categories is the study of certain kinds of colimits. These colimits are a generalization of coequalizers, and we shall see that they naturally appear in …
The study of codescent in Gray-categories is the study of certain kinds of colimits. These colimits are a generalization of coequalizers, and we shall see that they naturally appear in the study of algebras for Gray-monads. The Gray-category which represents codescent diagrams is here denoted ΔG for its close connection with the simplicial category. In fact, the underlying category of ΔG is the free category on the subcategory of Δop with objects [0], [1], [2],[3], and morphisms consisting of all face maps together with all degeneracy maps whose source is [0] or [1]. The definition given below actually uses the objects [1], [2], [3], [4], identifying this category with a subcategory of Δ instead (and here one should use the algebraist's Δ which includes the empty ordinal [0]). This presentation is not coincidental, as ΔG should be viewed as a kind of three-dimensional version of Δ which has the universal property of being the free strict monoidal category generated by a monoid. While we do not pursue this perspective any further, computing the higher dimensional analogues of the strict monoidal category Δ is an interesting open problem.
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In this chapter, we will prove a coherence theorem comparing free tricategories to free Gray-categories. This theorem states that the natural functor induced by the universal property from the free …
In this chapter, we will prove a coherence theorem comparing free tricategories to free Gray-categories. This theorem states that the natural functor induced by the universal property from the free tricategory to the free Gray-category on the same underlying data is a triequivalence. It is also a simple matter to prove a similar result comparing Gray-categories and strict 3-categories: the natural functor induced by the universal property from the free Gray-category to the free strict 3-category on the same underlying data is a triequivalence. This latter result might seem surprising, as it is well-known that not every tricategory is triequivalent to a strict 3-category, but in fact these results only express that the maps of monads from the free tricategory monad to the free Gray-category monad to the free strict 3-category monad can be equipped with contractions in the sense of Leinster (2004); this condition is one requirement for a monad to be a reasonable monad for a theory of weak 3-categories. As in the case of the coherence theory for bicategories, we can use this result to prove that diagrams of constraint 3-cells of a certain type always commute. Our results differ from the analogous ones for bicategories in that only some diagrams commute for tricategories but all diagrams of constraint 2-cells commute in a bicategory. As an example, we explicitly construct a diagram of constraint 3-cells that is not required to commute in general, and in fact does not commute in example tricategories which arise from braided monoidal categories.
This chapter will prove that every tricategory is triequivalent to a Gray-category by a Yoneda-style argument. Such a proof proceeds in a number of steps. First, we must study functor …
This chapter will prove that every tricategory is triequivalent to a Gray-category by a Yoneda-style argument. Such a proof proceeds in a number of steps. First, we must study functor tricategories. Second, we must produce a Yoneda embedding, and prove that it is actually an embedding. Finally, we must identify a sub-object of the target of the Yoneda embedding as the desired triequivalent Gray-category. In the case of coherence for bicategories, these were the only steps required; here we require one more initial step, namely that our tricategory T gets replaced by a cubical one.
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In this chapter, we will establish an important relationship between categories enriched over the monoidal category Gray and certain kinds of semi-strict tricategories. The first step is to define an …
In this chapter, we will establish an important relationship between categories enriched over the monoidal category Gray and certain kinds of semi-strict tricategories. The first step is to define an intermediate notion, that of a cubical tricategory. We will then show that strict, cubical tricategories are essentially Gray-categories. With this relationship in mind, we will then prove a weak form of coherence that will be necessary later, namely that every tricategory is triequivalent to a cubical one. This intermediate theorem appears in Gordon et al. (1995), and the presentation here follows that one closely. Finally, we will show that the canonical strictification B → stB for bicategories extends to a functor of tricategories st: Bicat → Gray.
In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that …
In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers. This is a version of Quillen's Theorem B amenable to applications. Second, we compute the $E^2$ page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of $S^{-1}S$. We show that if $S$ is a symmetric monoidal 2-groupoid with faithful translations then $S^{-1}S$ models the group completion of $S$.
This chapter will be a basic introduction to the theory of Gray-categories. There are a variety of natural ways to motivate the Gray-tensor product of 2-categories, and I would like …
This chapter will be a basic introduction to the theory of Gray-categories. There are a variety of natural ways to motivate the Gray-tensor product of 2-categories, and I would like to mention a few of them briefly without worrying about proofs of the various technical results that make this theory work. To be clear, I do not believe any of the material in this chapter is new; I have only collected together material on the Gray tensor product and Gray-categories that we will need later in studying either coherence for tricategories or the general coherence problem for algebras over Gray-monads. The main references are Gray's (1974, 1976) work, although the handwritten notes of Street (1988) provide another perspective. I have also drawn heavily from the material in Gordon–Power–and Street (1995). I do not know of a reference for the explanation of the Gray-tensor product in terms of a factorization, although it is mentioned in passing by Lack (2010b), and it was certainly from the lectures upon which that article is based that I learned that the Gray-tensor product could be expressed in this way.
This chapter will be devoted to studying some aspects of the total algebraic structure consisting of tricategories, functors, transformations, modifications, and perturbations. This chapter will only establish some basic properties …
This chapter will be devoted to studying some aspects of the total algebraic structure consisting of tricategories, functors, transformations, modifications, and perturbations. This chapter will only establish some basic properties that will be used later. There should be a weak 4-category Tricat, but constructing the entire structure would involve a substantial investment, much of which we will not need for the purposes of proving coherence. Since we will be proving a version of the Yoneda lemma for cubical tricategories, we will need to construct functor tricategories of the form [Top, Gray]. This functor tricategory would be the hom-tricategory in the putative construction of Tricat, but we only construct this in the special case when the target is a Gray-category, and this restriction greatly simplifies many of the calculations. The Yoneda embedding for cubical tricategories will be constructed in Chapter 9; for now we will focus on some basic composition formulas that will be required later.
This chapter will develop the basic tools necessary to construct free tricategories and free Gray-categories. First we must decide on the underlying data from which a tricategory is to be …
This chapter will develop the basic tools necessary to construct free tricategories and free Gray-categories. First we must decide on the underlying data from which a tricategory is to be generated freely. Second, we must construct both the free tricategory and the free Gray-category on this data. This requires a bit of care as one must pay careful attention to how the universal property is stated; the issue here is that, as we will see, the category of tricategories and strict maps has to be constructed directly, and not as a sub-object of some structure involving more general 1-cells. Finally, we prove some results analogous to those leading up to the proof of the coherence theorem for bicategories.
In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that …
In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers. This is a version of Quillen's Theorem B amenable to applications. Second, we compute the $E^2$ page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of $S^{-1}S$. We show that if $S$ is a symmetric monoidal 2-groupoid with faithful translations then $S^{-1}S$ models the group completion of $S$.
We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is …
We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric then this process can be iterated. We show that starting from the symmetric monoidal bicategory Cat and performing the construction twice yields a convenient symmetric monoidal bicategory of partially strict tricategories. We show that restricting to the doubly degenerate ones immediately gives the correct bicategory of "2-tuply monoidal categories" missing from our earlier studies of the Periodic Table. We propose a generalisation to all k-tuply monoidal n-categories.
We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpretted topologically …
We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpretted topologically using up-to-homotopy operad actions and the algebraic classification of surface braids.
We define a tensor product for permutative categories and prove a number of key properties. We show that this product makes the 2-category of permutative categories closed symmetric monoidal as …
We define a tensor product for permutative categories and prove a number of key properties. We show that this product makes the 2-category of permutative categories closed symmetric monoidal as a bicategory.
This work introduces a general theory of universal pseudomorphisms and develops their connection to diagrammatic coherence. The main results give hypotheses under which pseudomorphism coherence is equivalent to the coherence …
This work introduces a general theory of universal pseudomorphisms and develops their connection to diagrammatic coherence. The main results give hypotheses under which pseudomorphism coherence is equivalent to the coherence theory of strict algebras. Applications include diagrammatic coherence for plain, symmetric, and braided monoidal functors. The final sections include a variety of examples.
This work introduces a general theory of universal pseudomorphisms and develops their connection to diagrammatic coherence. The main results give hypotheses under which pseudomorphism coherence is equivalent to the coherence …
This work introduces a general theory of universal pseudomorphisms and develops their connection to diagrammatic coherence. The main results give hypotheses under which pseudomorphism coherence is equivalent to the coherence theory of strict algebras. Applications include diagrammatic coherence for plain, symmetric, and braided monoidal functors. The final sections include a variety of examples.
We define a tensor product for permutative categories and prove a number of key properties. We show that this product makes the 2-category of permutative categories closed symmetric monoidal as …
We define a tensor product for permutative categories and prove a number of key properties. We show that this product makes the 2-category of permutative categories closed symmetric monoidal as a bicategory.
In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that …
In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers. This is a version of Quillen's Theorem B amenable to applications. Second, we compute the $E^2$ page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of $S^{-1}S$. We show that if $S$ is a symmetric monoidal 2-groupoid with faithful translations then $S^{-1}S$ models the group completion of $S$.
In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that …
In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers. This is a version of Quillen's Theorem B amenable to applications. Second, we compute the $E^2$ page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of $S^{-1}S$. We show that if $S$ is a symmetric monoidal 2-groupoid with faithful translations then $S^{-1}S$ models the group completion of $S$.
Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1and 2-cells.The classifying space of a Picard 2-category D is an infinite loop space, the zeroth space of the K -theory …
Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1and 2-cells.The classifying space of a Picard 2-category D is an infinite loop space, the zeroth space of the K -theory spectrum KD.This spectrum has stable homotopy groups concentrated in levels 0, 1 and 2. We describe part of the Postnikov data of KD in terms of categorical structure.We use this to show that there is no strict skeletal Picard 2-category whose K -theory realizes the 2-truncation of the sphere spectrum.As part of the proof, we construct a categorical suspension, producing a Picard 2-category †C from a Picard 1-category C , and show that it commutes with K -theory, in that K †C is stably equivalent to †KC .
Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structurepreserving maps, but often the maps of interest are merely lax monoidal.We describe conditions under which …
Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structurepreserving maps, but often the maps of interest are merely lax monoidal.We describe conditions under which one can transport the weak equivalences from one category to another with the same objects and a broader class of maps.Under mild hypotheses this process produces an equivalence of homotopy theories.We describe examples including algebras over an operad, such as symmetric monoidal categories and n-fold monoidal categories; and diagram categories, such as Γ-categories.
We show that every action operad gives rise to a notion of monoidal category via the categorical version of the Borel construction, embedding action operads into the category of 2-monads …
We show that every action operad gives rise to a notion of monoidal category via the categorical version of the Borel construction, embedding action operads into the category of 2-monads on $\mathbf{Cat}$. We characterize those 2-monads in the image of this embedding, and as an example show that the theory of coboundary categories corresponds precisely to the operad of $n$-fruit cactus groups. We finally define $\mathbf{\Lambda}$-multicategories for an action operad $\mathbf{\Lambda}$, and show that they arise as monads in a Kleisli bicategory.
Constructions of spaces from symmetric monoidal categories or $\Gamma$-categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal or lax …
Constructions of spaces from symmetric monoidal categories or $\Gamma$-categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal or lax natural. We describe conditions under which one can transport the weak equivalences from such a category of structured objects and strict maps to the corresponding category with the same objects and lax maps. Under mild hypotheses this process produces an equivalence of homotopy theories. We describe examples including algebras over an operad, such as symmetric monoidal categories and $n$-fold monoidal categories; and diagram categories, such as $\Gamma$-categories.
It was argued by Crans that it is too much to ask that the category of Gray-categories admit a well behaved monoidal biclosed structure. We make this precise by establishing …
It was argued by Crans that it is too much to ask that the category of Gray-categories admit a well behaved monoidal biclosed structure. We make this precise by establishing undesirable properties that any such monoidal biclosed structure must have. In particular we show that there does not exist any tensor product making the model category of Gray-categories into a monoidal model category.
We show that every action operad gives rise to a notion of monoidal category via the categorical version of the Borel construction, embedding action operads into the category of 2-monads …
We show that every action operad gives rise to a notion of monoidal category via the categorical version of the Borel construction, embedding action operads into the category of 2-monads on $\mathbf{Cat}$. We characterize those 2-monads in the image of this embedding, and as an example show that the theory of coboundary categories corresponds precisely to the operad of $n$-fruit cactus groups. We finally define $\mathbf{\Lambda}$-multicategories for an action operad $\mathbf{\Lambda}$, and show that they arise as monads in a Kleisli bicategory.
Abstract A multivariable adjunction is the generalisation of the notion of a 2-variable adjunction, the classical example being the hom/tensor/cotensor trio of functors, to n + 1 functors of n …
Abstract A multivariable adjunction is the generalisation of the notion of a 2-variable adjunction, the classical example being the hom/tensor/cotensor trio of functors, to n + 1 functors of n variables. In the presence of multivariable adjunctions, natural transformations between certain composites built from multivariable functors have “dual” forms. We refer to corresponding natural transformations as multivariable or parametrised mates, generalising the mates correspondence for ordinary adjunctions, which enables one to pass between natural transformations involving left adjoints to those involving right adjoints. A central problem is how to express the naturality (or functoriality) of the parametrised mates, giving a precise characterization of the dualities so-encoded. We present the notion of “cyclic double multicategory” as a structure in which to organise multivariable adjunctions and mates. While the standard mates correspondence is described using an isomorphism of double categories, the multivariable version requires the framework of “double multicategories”. Moreover, we show that the analogous isomorphisms of double multicategories give a cyclic action on the multimaps, yielding the notion of “cyclic double multicategory”. The work is motivated by and applied to Riehl's approach to algebraic monoidal model categories.
It was argued by Crans that it is too much to ask that the category of Gray-categories admit a well behaved monoidal biclosed structure. We make this precise by establishing …
It was argued by Crans that it is too much to ask that the category of Gray-categories admit a well behaved monoidal biclosed structure. We make this precise by establishing undesirable properties that any such monoidal biclosed structure must have. In particular we show that there does not exist any tensor product making the model category of Gray-categories into a monoidal model category.
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study …
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study the 2-category of algebras over an operad in Cat. We show that any operad is finitary, that an operad is cartesian if and only if the group actions are nearly free (in a precise fashion), and that the existence of a pseudo-commutative structure largely depends on the groups of equivariance. We conclude by showing that the operad for strict braided monoidal categories has two canonical pseudo-commutative structures.
We study the totality of weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric …
We study the totality of weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric then this process can be iterated. We show that starting from the symmetric monoidal bicategory Cat and performing the construction twice yields a convenient symmetric monoidal bicategory of partially strict tricategories. We show that restricting to the doubly degenerate ones immediately gives the correct bicategory of 2-tuply monoidal categories missing from our earlier studies of the Periodic Table. We propose a generalisation to all k-tuply monoidal n-categories.
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The study of codescent in Gray-categories is the study of certain kinds of colimits. These colimits are a generalization of coequalizers, and we shall see that they naturally appear in …
The study of codescent in Gray-categories is the study of certain kinds of colimits. These colimits are a generalization of coequalizers, and we shall see that they naturally appear in the study of algebras for Gray-monads. The Gray-category which represents codescent diagrams is here denoted ΔG for its close connection with the simplicial category. In fact, the underlying category of ΔG is the free category on the subcategory of Δop with objects [0], [1], [2],[3], and morphisms consisting of all face maps together with all degeneracy maps whose source is [0] or [1]. The definition given below actually uses the objects [1], [2], [3], [4], identifying this category with a subcategory of Δ instead (and here one should use the algebraist's Δ which includes the empty ordinal [0]). This presentation is not coincidental, as ΔG should be viewed as a kind of three-dimensional version of Δ which has the universal property of being the free strict monoidal category generated by a monoid. While we do not pursue this perspective any further, computing the higher dimensional analogues of the strict monoidal category Δ is an interesting open problem.
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In this chapter, we will prove a coherence theorem comparing free tricategories to free Gray-categories. This theorem states that the natural functor induced by the universal property from the free …
In this chapter, we will prove a coherence theorem comparing free tricategories to free Gray-categories. This theorem states that the natural functor induced by the universal property from the free tricategory to the free Gray-category on the same underlying data is a triequivalence. It is also a simple matter to prove a similar result comparing Gray-categories and strict 3-categories: the natural functor induced by the universal property from the free Gray-category to the free strict 3-category on the same underlying data is a triequivalence. This latter result might seem surprising, as it is well-known that not every tricategory is triequivalent to a strict 3-category, but in fact these results only express that the maps of monads from the free tricategory monad to the free Gray-category monad to the free strict 3-category monad can be equipped with contractions in the sense of Leinster (2004); this condition is one requirement for a monad to be a reasonable monad for a theory of weak 3-categories. As in the case of the coherence theory for bicategories, we can use this result to prove that diagrams of constraint 3-cells of a certain type always commute. Our results differ from the analogous ones for bicategories in that only some diagrams commute for tricategories but all diagrams of constraint 2-cells commute in a bicategory. As an example, we explicitly construct a diagram of constraint 3-cells that is not required to commute in general, and in fact does not commute in example tricategories which arise from braided monoidal categories.
This chapter will prove that every tricategory is triequivalent to a Gray-category by a Yoneda-style argument. Such a proof proceeds in a number of steps. First, we must study functor …
This chapter will prove that every tricategory is triequivalent to a Gray-category by a Yoneda-style argument. Such a proof proceeds in a number of steps. First, we must study functor tricategories. Second, we must produce a Yoneda embedding, and prove that it is actually an embedding. Finally, we must identify a sub-object of the target of the Yoneda embedding as the desired triequivalent Gray-category. In the case of coherence for bicategories, these were the only steps required; here we require one more initial step, namely that our tricategory T gets replaced by a cubical one.
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In this chapter, we will establish an important relationship between categories enriched over the monoidal category Gray and certain kinds of semi-strict tricategories. The first step is to define an …
In this chapter, we will establish an important relationship between categories enriched over the monoidal category Gray and certain kinds of semi-strict tricategories. The first step is to define an intermediate notion, that of a cubical tricategory. We will then show that strict, cubical tricategories are essentially Gray-categories. With this relationship in mind, we will then prove a weak form of coherence that will be necessary later, namely that every tricategory is triequivalent to a cubical one. This intermediate theorem appears in Gordon et al. (1995), and the presentation here follows that one closely. Finally, we will show that the canonical strictification B → stB for bicategories extends to a functor of tricategories st: Bicat → Gray.
This chapter will be a basic introduction to the theory of Gray-categories. There are a variety of natural ways to motivate the Gray-tensor product of 2-categories, and I would like …
This chapter will be a basic introduction to the theory of Gray-categories. There are a variety of natural ways to motivate the Gray-tensor product of 2-categories, and I would like to mention a few of them briefly without worrying about proofs of the various technical results that make this theory work. To be clear, I do not believe any of the material in this chapter is new; I have only collected together material on the Gray tensor product and Gray-categories that we will need later in studying either coherence for tricategories or the general coherence problem for algebras over Gray-monads. The main references are Gray's (1974, 1976) work, although the handwritten notes of Street (1988) provide another perspective. I have also drawn heavily from the material in Gordon–Power–and Street (1995). I do not know of a reference for the explanation of the Gray-tensor product in terms of a factorization, although it is mentioned in passing by Lack (2010b), and it was certainly from the lectures upon which that article is based that I learned that the Gray-tensor product could be expressed in this way.
This chapter will be devoted to studying some aspects of the total algebraic structure consisting of tricategories, functors, transformations, modifications, and perturbations. This chapter will only establish some basic properties …
This chapter will be devoted to studying some aspects of the total algebraic structure consisting of tricategories, functors, transformations, modifications, and perturbations. This chapter will only establish some basic properties that will be used later. There should be a weak 4-category Tricat, but constructing the entire structure would involve a substantial investment, much of which we will not need for the purposes of proving coherence. Since we will be proving a version of the Yoneda lemma for cubical tricategories, we will need to construct functor tricategories of the form [Top, Gray]. This functor tricategory would be the hom-tricategory in the putative construction of Tricat, but we only construct this in the special case when the target is a Gray-category, and this restriction greatly simplifies many of the calculations. The Yoneda embedding for cubical tricategories will be constructed in Chapter 9; for now we will focus on some basic composition formulas that will be required later.
This chapter will develop the basic tools necessary to construct free tricategories and free Gray-categories. First we must decide on the underlying data from which a tricategory is to be …
This chapter will develop the basic tools necessary to construct free tricategories and free Gray-categories. First we must decide on the underlying data from which a tricategory is to be generated freely. Second, we must construct both the free tricategory and the free Gray-category on this data. This requires a bit of care as one must pay careful attention to how the universal property is stated; the issue here is that, as we will see, the category of tricategories and strict maps has to be constructed directly, and not as a sub-object of some structure involving more general 1-cells. Finally, we prove some results analogous to those leading up to the proof of the coherence theorem for bicategories.
In the study of higher categories, dimension three occupies an interesting position on the landscape of higher dimensional category theory. From the perspective of a “hands-on” approach to defining weak …
In the study of higher categories, dimension three occupies an interesting position on the landscape of higher dimensional category theory. From the perspective of a “hands-on” approach to defining weak n-categories, tricategories represent the most complicated kind of higher category that the community at large seems comfortable working with. On the other hand, dimension three is the lowest dimension in which strict n-categories are genuinely more restrictive than fully weak ones, so tricategories should be a sort of jumping off point for understanding general higher dimensional phenomena. This work is intended to provide an accessible introduction to coherence problems in three-dimensional category.
We study the monoidal structure of the standard strictification functor $\textrm{st}:\mathbf{Bicat} \rightarrow \mathbf{2Cat}$. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the …
We study the monoidal structure of the standard strictification functor $\textrm{st}:\mathbf{Bicat} \rightarrow \mathbf{2Cat}$. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the 2-category whose objects are 2-categories.
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study …
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study the 2-category of algebras over an operad in Cat. We show that any operad is finitary, that an operad is cartesian if and only if the group actions are nearly free (in a precise fashion), and that the existence of a pseudo-commutative structure largely depends on the groups of equivariance. We conclude by showing that the operad for strict braided monoidal categories has two canonical pseudo-commutative structures.
We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is …
We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric then this process can be iterated. We show that starting from the symmetric monoidal bicategory Cat and performing the construction twice yields a convenient symmetric monoidal bicategory of partially strict tricategories. We show that restricting to the doubly degenerate ones immediately gives the correct bicategory of "2-tuply monoidal categories" missing from our earlier studies of the Periodic Table. We propose a generalisation to all k-tuply monoidal n-categories.
We present the notion of "cyclic double multicategory", as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2-variable adjunction is the hom/tensor/cotensor trio …
We present the notion of "cyclic double multicategory", as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2-variable adjunction is the hom/tensor/cotensor trio of functors; we generalise this situation to n+1 functors of n variables. Furthermore, we generalise the mates correspondence, which enables us to pass between natural transformations involving left adjoints to those involving right adjoints. While the standard mates correspondence is described using an isomorphism of double categories, the multivariable version requires the framework of "double multicategories". Moreover, we show that the analogous isomorphisms of double multicategories give a cyclic action on the multimaps, yielding the notion of "cyclic double multicategory". The work is motivated by and applied to Riehl's approach to algebraic monoidal model categories.
We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpretted topologically …
We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpretted topologically using up-to-homotopy operad actions and the algebraic classification of surface braids.
We show that every internal biequivalence in a tricategory T is part of a biadjoint biequivalence. We give two applications of this result, one for transporting monoidal structures and one …
We show that every internal biequivalence in a tricategory T is part of a biadjoint biequivalence. We give two applications of this result, one for transporting monoidal structures and one for equipping a monoidal bicategory with invertible objects with a coherent choice of those inverses.
Nous continuons le travail commence en [5] en etudiant les tricategories degenerees et en les comparant avec les structures predites par le tableau periodique des n-categories. Pour les tricategories trois …
Nous continuons le travail commence en [5] en etudiant les tricategories degenerees et en les comparant avec les structures predites par le tableau periodique des n-categories. Pour les tricategories trois fois degenerees nous demontrons une triequivalence avec la tricategorie partiellement discrete des monoides commutatifs. Pour les tricategories deux fois degenerees nous expliquons comment on peut construire une categorie monoidale tressee d'une tricategorie deux fois degeneree donnee, mais nous demontrons que cette construction n'induit pas une comparaison simple entre BrMonCat et Tricat. Nous discutons comment on peut iterer la construction des icones pour produire un equivalence, mais nous esperons a la suite pour donner les details. Finalement nous etudions les tricategories degenerees pour donner la premiere definition de bicategorie monoidale completement algebrique et la structure entiere de tricategorie de MonBicat. We continue the project begun in [5] by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. For triply degenerate tricategories we exhibit a triequivalence with the partially discrete tricategory of commutative monoids. For the doubly degenerate case we explain how to construct a braided monoidal category from a given doubly degenerate category, but show that this does not induce a straightforward comparison between BrMonCat and Tricat. We indicate how to iterate the icon construction to produce an equivalence, but leave the details to a sequel. Finally we study degenerate tricategories in order to give the first fully algebraic definition of monoidal bicategories and the full tricategory structure MonBicat.
We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpretted topologically …
We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpretted topologically using up-to-homotopy operad actions and the algebraic classification of surface braids.
We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called …
We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical bicategory, this being a bicategory enriched in the monoidal 2-category of pseudo double categories. Finally, we show that every sufficiently well-behaved locally cubical bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.
We form tricategories and the homomorphisms between them into a bicategory. We then enrich this bicategory into an example of a three-dimensional structure called a locally double bicategory, this being …
We form tricategories and the homomorphisms between them into a bicategory. We then enrich this bicategory into an example of a three-dimensional structure called a locally double bicategory, this being a bicategory enriched in the monoidal 2-category of weak double categories. Finally, we show that every sufficiently well-behaved locally double bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.
We continue the project begun in ``The periodic table of $n$-categories for low dimensions I'' by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. …
We continue the project begun in ``The periodic table of $n$-categories for low dimensions I'' by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. For triply degenerate tricategories we exhibit a triequivalence with the partially discrete tricategory of commutative monoids. For the doubly degenerate case we explain how to construct a braided monoidal category from a given doubly degenerate category, but show that this does not induce a straightforward comparison between \bfseries{BrMonCat} and \bfseries{Tricat}. We show how to alter the natural structure of \bfseries{Tricat} in two different ways to provide a comparison, but show that only the more brutal alteration yields an equivalence. Finally we study degenerate tricategories in order to give the first fully algebraic definition of monoidal bicategories and the full tricategory structure \bfseries{MonBicat}.
We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and …
We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and degenerate bicategories to monoidal categories; however, to understand this correspondence fully we examine the totalities of such structures together with maps between them and higher maps between those. Categories naturally form a 2-category {\bfseries Cat} so we take the full sub-2-category of this whose 0-cells are the degenerate categories. Monoids naturally form a category, but we regard this as a discrete 2-category to make the comparison. We show that this construction does not yield a biequivalence; to get an equivalence we ignore the natural transformations and consider only the {\it category} of degenerate categories. A similar situation occurs for degenerate bicategories. The tricategory of such does not yield an equivalence with monoidal categories; we must consider only the categories of such structures. For doubly degenerate bicategories the tricategory of such is not naturally triequivalent to the category of commutative monoids (regarded as a tricategory). However in this case considering just the categories does not give an equivalence either; to get an equivalence we consider the {\it bicategory} of doubly degenerate bicategories. We conclude with a hypothesis about how the above cases might generalise for n-fold degenerate n-categories.
We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and …
We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and degenerate bicategories to monoidal categories; however, to understand this correspondence fully we examine the totalities of such structures together with maps between them and higher maps between those. Categories naturally form a 2-category {\bfseries Cat} so we take the full sub-2-category of this whose 0-cells are the degenerate categories. Monoids naturally form a category, but we regard this as a discrete 2-category to make the comparison. We show that this construction does not yield a biequivalence; to get an equivalence we ignore the natural transformations and consider only the {\it category} of degenerate categories. A similar situation occurs for degenerate bicategories. The tricategory of such does not yield an equivalence with monoidal categories; we must consider only the categories of such structures. For doubly degenerate bicategories the tricategory of such is not naturally triequivalent to the category of commutative monoids (regarded as a tricategory). However in this case considering just the categories does not give an equivalence either; to get an equivalence we consider the {\it bicategory} of doubly degenerate bicategories. We conclude with a hypothesis about how the above cases might generalise for n-fold degenerate n-categories.
We provide a complete generators and relations presentation of the 2-dimensional extended unoriented and oriented bordism bicategories as symmetric monoidal bicategories. Thereby we classify these types of 2-dimensional extended topological …
We provide a complete generators and relations presentation of the 2-dimensional extended unoriented and oriented bordism bicategories as symmetric monoidal bicategories. Thereby we classify these types of 2-dimensional extended topological field theories with arbitrary target bicategory. As an immediate corollary we obtain a concrete classification when the target is the symmetric monoidal bicategory of algebras, bimodules, and intertwiners over a fixed commutative ground ring. In the oriented case, such an extended topological field theory is equivalent to specifying a (non-commutative) separable symmetric Frobenius algebra. The text is divided into three chapters. The first develops a variant of higher Morse theory and uses it to obtain a combinatorial description of surfaces suitable for the higher categorical language used later. The second chapter is an extensive treatment of the theory of symmetric monoidal bicategories. We introduce several stricter variants on the notion of symmetric monoidal bicategory, and give a very general treatment of the notion of presentation by generators and relations. Finally we provide a host of strictification and cohernece results for symmetric monoidal bicategories. The final chapter focuses on extended tqfts. We give a precise treatment of the extended bordism bicategory equipped with additional structure (such as framings or orientations). We apply the results of the previous two chapters to obtain a simple presentation of both the oriented and unoriented bordism bicategories, and describe the general method to obtain such classifications for other choices of structure. We examine the consequences of our classification when the target is the bicategory of algebras, bimodules, and maps, over a fixed commutative ground ring.
Abstract A Quillen model structure on the category Gray-Cat of Gray -categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full …
Abstract A Quillen model structure on the category Gray-Cat of Gray -categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full subcategory Gray-Gpd of Gray -groupoids. This is used to provide a functorial and model-theoretic proof of the unpublished theorem of Joyal and Tierney that Gray -groupoids model homotopy 3-types. The model structure on Gray-Cat is conjectured to be Quillen equivalent to a model structure on the category Tricat of tricategories and strict homomorphisms of tricategories.
Classication of homotopy n-types has focused on developing algebraic categories which are equivalent to categories of n-types. We expand this theory by providing algebraic models of homotopy-theoretic constructions for stable …
Classication of homotopy n-types has focused on developing algebraic categories which are equivalent to categories of n-types. We expand this theory by providing algebraic models of homotopy-theoretic constructions for stable one-types. These include a model for the Postnikov one-truncation of the sphere spectrum, and for its action on the model of a stable one-type. We show that a bicategorical cokernel introduced by Vitale models the cober of a map between stable one-types, and apply this to develop an algebraic model for the Postnikov data of a stable one-type.
Classification of homotopy n-types has focused on developing algebraic categories which are equivalent to categories of n-types. We expand this theory by providing algebraic models of homotopy-theoretic constructions for stable …
Classification of homotopy n-types has focused on developing algebraic categories which are equivalent to categories of n-types. We expand this theory by providing algebraic models of homotopy-theoretic constructions for stable one-types. These include a model for the Postnikov one-truncation of the sphere spectrum, and for its action on the model of a stable one-type. We show that a bicategorical cokernel introduced by Vitale models the cofiber of a map between stable one-types, and apply this to develop an algebraic model for the Postnikov data of a stable one-type.
This paper explores the relationship amongst the various simplicial and pseudosimplicial objects characteristically associated to any bicategory C .It proves the fact that the geometric realizations of all of these …
This paper explores the relationship amongst the various simplicial and pseudosimplicial objects characteristically associated to any bicategory C .It proves the fact that the geometric realizations of all of these possible candidate "nerves of C " are homotopy equivalent.Any one of these realizations could therefore be taken as the classifying space BC of the bicategory.Its other major result proves a direct extension of Thomason's "Homotopy Colimit Theorem" to bicategories: When the homotopy colimit construction is carried out on a diagram of spaces obtained by applying the classifying space functor to a diagram of bicategories, the resulting space has the homotopy type of a certain bicategory, called the "Grothendieck construction on the diagram".Our results provide coherence for all reasonable extensions to bicategories of Quillen's definition of the "classifying space" of a category as the geometric realization of the category's Grothendieck nerve, and they are applied to monoidal (tensor) categories through the elemental "delooping" construction.
Nous continuons le travail commence en [5] en etudiant les tricategories degenerees et en les comparant avec les structures predites par le tableau periodique des n-categories. Pour les tricategories trois …
Nous continuons le travail commence en [5] en etudiant les tricategories degenerees et en les comparant avec les structures predites par le tableau periodique des n-categories. Pour les tricategories trois fois degenerees nous demontrons une triequivalence avec la tricategorie partiellement discrete des monoides commutatifs. Pour les tricategories deux fois degenerees nous expliquons comment on peut construire une categorie monoidale tressee d'une tricategorie deux fois degeneree donnee, mais nous demontrons que cette construction n'induit pas une comparaison simple entre BrMonCat et Tricat. Nous discutons comment on peut iterer la construction des icones pour produire un equivalence, mais nous esperons a la suite pour donner les details. Finalement nous etudions les tricategories degenerees pour donner la premiere definition de bicategorie monoidale completement algebrique et la structure entiere de tricategorie de MonBicat. We continue the project begun in [5] by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. For triply degenerate tricategories we exhibit a triequivalence with the partially discrete tricategory of commutative monoids. For the doubly degenerate case we explain how to construct a braided monoidal category from a given doubly degenerate category, but show that this does not induce a straightforward comparison between BrMonCat and Tricat. We indicate how to iterate the icon construction to produce an equivalence, but leave the details to a sequel. Finally we study degenerate tricategories in order to give the first fully algebraic definition of monoidal bicategories and the full tricategory structure MonBicat.
We show how to construct a Gamma-bicategory from a symmetric monoidal bicategory, and use that to show that the classifying space is an infinite loop space upon group completion. We …
We show how to construct a Gamma-bicategory from a symmetric monoidal bicategory, and use that to show that the classifying space is an infinite loop space upon group completion. We also show a way to relate this construction to the classic Gamma-category construction for a bipermutative category. As an example, we use this machinery to construct a delooping of the K-theory of a bimonoidal category as defined by Baas-Dundas-Rognes.
We show that every internal biequivalence in a tricategory T is part of a biadjoint biequivalence. We give two applications of this result, one for transporting monoidal structures and one …
We show that every internal biequivalence in a tricategory T is part of a biadjoint biequivalence. We give two applications of this result, one for transporting monoidal structures and one for equipping a monoidal bicategory with invertible objects with a coherent choice of those inverses.
Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. It draws its inspiration from areas as diverse as topology, quantum algebra, mathematical …
Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. It draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics, logic, and theoretical computer science. The heart of this book is the language of generalized operads. This is as natural and transparent a language for higher category theory as the language of sheaves is for algebraic geometry, or vector spaces for linear algebra. It is introduced carefully, then used to give simple descriptions of a variety of higher categorical structures. In particular, one possible definition of n-category is discussed in detail, and some common aspects of other possible definitions are established. This is the first book on the subject and lays its foundations. It will appeal to both graduate students and established researchers who wish to become acquainted with this modern branch of mathematics.
The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for …
The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations. First we describe a "suspension" operation on n-categories, and hypothesize that the k-fold suspension of a weak n-category stabilizes for k >= n+2. We give evidence for this hypothesis and describe its relation to stable homotopy theory. We then propose a description of n-dimensional unitary extended TQFTs as weak n-functors from the "free stable weak n-category with duals on one object" to the n-category of "n-Hilbert spaces". We conclude by describing n-categorical generalizations of deformation quantization and the quantum double construction.
We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy …
We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy theory”, or more precisely that the category of such models has a well-behaved internal hom-object.
Thomason showed that the K -theory of symmetric monoidal categories models all connective spectra. This paper describes a new construction of a permutative category from a \Gamma -space, which is …
Thomason showed that the K -theory of symmetric monoidal categories models all connective spectra. This paper describes a new construction of a permutative category from a \Gamma -space, which is then used to re-prove Thomason's theorem and a non-completed variant.
We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and …
We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and degenerate bicategories to monoidal categories; however, to understand this correspondence fully we examine the totalities of such structures together with maps between them and higher maps between those. Categories naturally form a 2-category {\bfseries Cat} so we take the full sub-2-category of this whose 0-cells are the degenerate categories. Monoids naturally form a category, but we regard this as a discrete 2-category to make the comparison. We show that this construction does not yield a biequivalence; to get an equivalence we ignore the natural transformations and consider only the {\it category} of degenerate categories. A similar situation occurs for degenerate bicategories. The tricategory of such does not yield an equivalence with monoidal categories; we must consider only the categories of such structures. For doubly degenerate bicategories the tricategory of such is not naturally triequivalent to the category of commutative monoids (regarded as a tricategory). However in this case considering just the categories does not give an equivalence either; to get an equivalence we consider the {\it bicategory} of doubly degenerate bicategories. We conclude with a hypothesis about how the above cases might generalise for n-fold degenerate n-categories.
A may bear many monoidal structures, but (to within a isomorphism) only one of category with finite products. To capture such distinctions, we consider on a 2-category those 2-monads for …
A may bear many monoidal structures, but (to within a isomorphism) only one of category with finite products. To capture such distinctions, we consider on a 2-category those 2-monads for which algebra is if it exists, giving a precise mathematical definition of essentially unique and investigating its consequences. We call such 2-monads property-like. We further consider the more restricted class of fully property-like 2-monads, consisting of those property-like 2-monads for which all 2-cells between (even lax) algebra morphisms are algebra 2-cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which structure is adjoint to unit, and which we now call lax-idempotent 2-monads: both these and their colax-idempotent duals are fully property-like. We end by showing that (at least for finitary 2-monads) the classes of property-likes, fully property-likes, and lax-idempotents are each coreflective among all 2-monads.
We introduce two coloured operads in sets - the lattice path op- erad and a cyclic extension of it - closely related to iterated loop spaces and to universal operations …
We introduce two coloured operads in sets - the lattice path op- erad and a cyclic extension of it - closely related to iterated loop spaces and to universal operations on cochains. As main application we present a for- mal construction of an E2-action (resp. framed E2-action) on the Hochschild cochain complex of an associative (resp. symmetric Frobenius) algebra.
Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic theory. With the development of Quillen's c
Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic theory. With the development of Quillen's c
As an example of the categorical apparatus of pseudo algebras over 2-theories, we show that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with …
As an example of the categorical apparatus of pseudo algebras over 2-theories, we show that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with folding or as appropriate 2-functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2-cells of the horizontal 2-category. As a special case, strict 2-algebras with one object and everything invertible are crossed modules under a group.
Various concerns suggest looking for internal co-categories in categories with strong logical structure. It turns out that in any coherent category, all co-categories are co-equivalence relations.
Various concerns suggest looking for internal co-categories in categories with strong logical structure. It turns out that in any coherent category, all co-categories are co-equivalence relations.
We study the monoidal structure of the standard strictification functor $\textrm{st}:\mathbf{Bicat} \rightarrow \mathbf{2Cat}$. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the …
We study the monoidal structure of the standard strictification functor $\textrm{st}:\mathbf{Bicat} \rightarrow \mathbf{2Cat}$. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the 2-category whose objects are 2-categories.
In this paper we answer the question: `what kind of a structure can a general multicategory be enriched in?' The answer is, in a sense to be made precise, that …
In this paper we answer the question: `what kind of a structure can a general multicategory be enriched in?' The answer is, in a sense to be made precise, that a multicategory of one type can be enriched in a multicategory of the type one level up. In the case of ordinary categories this reduces to something surprising: a category may be enriched in an `fc-multicategory', a very general kind of 2-dimensional structure encompassing monoidal categories, plain multicategories, bicategories and double categories. (It turns out that fc-multicategories also provide a natural setting for the bimodules construction.) An extended application is given: the relaxed multicategories of Borcherds and Soibelman are explained in terms of enrichment.
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study …
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study the 2-category of algebras over an operad in Cat. We show that any operad is finitary, that an operad is cartesian if and only if the group actions are nearly free (in a precise fashion), and that the existence of a pseudo-commutative structure largely depends on the groups of equivariance. We conclude by showing that the operad for strict braided monoidal categories has two canonical pseudo-commutative structures.
The work to be presented in this paper has been inspired by several of Professor Graeme Segal's papers. Our search for a geometrically defined elliptic cohomology theory with associated elliptic …
The work to be presented in this paper has been inspired by several of Professor Graeme Segal's papers. Our search for a geometrically defined elliptic cohomology theory with associated elliptic objects obviously stems from his Bourbaki seminar. Our readiness to form group completions of symmetric monoidal categories by passage to algebraic K-theory spectra derives from his Topology paper. Our inclination to invoke 2-functors to the 2-category of 2-vector spaces generalizes his model for topological K-theory in terms of functors from a path category to the category of vector spaces. We offer him our admiration.
We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given monoidal model category. This yields a unified treatment for the known …
We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given monoidal model category. This yields a unified treatment for the known model structures on simplicial, topological, dg- and spectral categories. Our proof is mainly based on a fundamental property of cofibrant enriched categories on two objects, stated below as the Interval Cofibrancy Theorem.
Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery …
Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal in this paper is expository: we explain both approaches and a proof of their equivalence. Our second goal is to generalize this result to enriched categories and homotopy weighted limits, showing that the classical explicit constructions still give the right answer in the abstract sense. This result partially bridges the gap between classical homotopy theory and modern abstract homotopy theory. To do this we introduce a notion of "enriched homotopical categories", which are more general than enriched model categories, but are still a good place to do enriched homotopy theory. This demonstrates that the presence of enrichment often simplifies rather than complicates matters, and goes some way toward achieving a better understanding of "the role of homotopy in homotopy theory."
The notion of cartesian bicategory, introduced by Carboni and Walters for locally ordered bicategories, is extended to general bicategories. It is shown that a cartesian bicategory is a symmetric monoidal …
The notion of cartesian bicategory, introduced by Carboni and Walters for locally ordered bicategories, is extended to general bicategories. It is shown that a cartesian bicategory is a symmetric monoidal bicategory.