We provide a complete description of the category of pseudo-categories (including pseudo-functors, natural and pseudo-natural transformations and pseudo modifications). A pseudo-category is a non strict version of an internal category. …
We provide a complete description of the category of pseudo-categories (including pseudo-functors, natural and pseudo-natural transformations and pseudo modifications). A pseudo-category is a non strict version of an internal category. It was called a weak category and weak double category in some earlier papers. When internal to Cat it is at the same time a generalization of a bicategory and a double category.
We introduce pseudoalgebras for relative pseudomonads and develop their theory. For each relative pseudomonad $T$, we construct a free--forgetful relative pseudoadjunction that exhibits the bicategory of $T$-pseudoalgebras as terminal among …
We introduce pseudoalgebras for relative pseudomonads and develop their theory. For each relative pseudomonad $T$, we construct a free--forgetful relative pseudoadjunction that exhibits the bicategory of $T$-pseudoalgebras as terminal among resolutions of $T$. The Kleisli bicategory for $T$ thus embeds into the bicategory of pseudoalgebras as the sub-bicategory of free pseudoalgebras. We consequently obtain a coherence theorem that implies, for instance, that the bicategory of distributors is biequivalent to the 2-category of presheaf categories. In doing so, we extend several aspects of the theory of pseudomonads to relative pseudomonads, including doctrinal adjunction, transport of structure, and lax-idempotence. As an application of our general theory, we prove that, for each class of colimits $\Phi$, there is a correspondence between monads relative to free $\Phi$-cocompletions, and $\Phi$-cocontinuous monads on free $\Phi$-cocompletions.
In the introduction to [3] it is mistakenly claimed that Mislin's proof uses Carlsson's proof of Segal's Burnside ring conjecture. In fact, Mislin uses instead the work of Carlsson, Miller …
In the introduction to [3] it is mistakenly claimed that Mislin's proof uses Carlsson's proof of Segal's Burnside ring conjecture. In fact, Mislin uses instead the work of Carlsson, Miller and Lannes on Sullivan's fixed-point conjecture, and work of Dwyer and Zabrodski [1]. There is, however, a proof by Snaith [2] that depends on a version of the Segal conjecture. 2000 Mathematics Subject Classification 20J06 (primary), 20C05 (secondary).
We present Trimble's definition of a tetracategory and prove that the spans in (strict) 2-categories with certain limits have the structure of a monoidal tricategory, defined as a one-object tetracategory. …
We present Trimble's definition of a tetracategory and prove that the spans in (strict) 2-categories with certain limits have the structure of a monoidal tricategory, defined as a one-object tetracategory. We recall some notions of limits in 2-categories for use in the construction of the monoidal tricategory of spans.
We present Trimble's definition of a tetracategory and prove that the spans in (strict) 2-categories with certain limits have the structure of a monoidal tricategory, defined as a one-object tetracategory. …
We present Trimble's definition of a tetracategory and prove that the spans in (strict) 2-categories with certain limits have the structure of a monoidal tricategory, defined as a one-object tetracategory. We recall some notions of limits in 2-categories for use in the construction of the monoidal tricategory of spans.
This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and …
This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves.
Reflective Subcategories, Localizations and Factorization Systems: Corrigenda C. Cassidy, M. Hébert and G. M. Kelly, 1980 Mathematics subject classification (Amer. Math. Soc.): 18 A 20.
Reflective Subcategories, Localizations and Factorization Systems: Corrigenda C. Cassidy, M. Hébert and G. M. Kelly, 1980 Mathematics subject classification (Amer. Math. Soc.): 18 A 20.
We introduce the notion of a relative pseudomonad, which generalizes the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient …
We introduce the notion of a relative pseudomonad, which generalizes the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonads on the Kleisli bicategory of a relative pseudomonad. The results are applied to define several pseudomonads on the bicategory of profunctors in an homogeneous way and provide a uniform approach to the definition of bicategories that are of interest in operad theory, mathematical logic, and theoretical computer science.
Given a full subcategory [Fscr ] of a category [Ascr ] , the existence of left [Fscr ] -approximations (or [Fscr ] -preenvelopes) completing diagrams in a unique way is …
Given a full subcategory [Fscr ] of a category [Ascr ] , the existence of left [Fscr ] -approximations (or [Fscr ] -preenvelopes) completing diagrams in a unique way is equivalent to the fact that [Fscr ] is reflective in [Ascr ] , in the classical terminology of category theory. In the first part of the paper we establish, for a rather general [Ascr ] , the relationship between reflectivity and covariant finiteness of [Fscr ] in [Ascr ] , and generalize Freyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflections with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories. In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres ( M ) and Add ( M ). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcategories of {\rm Mod}_{R} , and has some connections with the “pure-semisimple conjecture”. 1991 Mathematics Subject Classification 18A40, 16D90, 16E70.
If $f:S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal functor $f_\otimes: \mathcal H_*(S') \to\mathcal H_*(S)$, where $\mathcal H_*(S)$ is the pointed unstable …
If $f:S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal functor $f_\otimes: \mathcal H_*(S') \to\mathcal H_*(S)$, where $\mathcal H_*(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite etale, we show that it stabilizes to a functor $f_\otimes: \mathcal{SH}(S') \to \mathcal{SH}(S)$, where $\mathcal{SH}(S)$ is the $\mathbb P^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $H\mathbb Z$, the homotopy K-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb Z$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.
This article introduces Globular, an online proof assistant for the formalization and verification of proofs in higher-dimensional category theory. The tool produces graphical visualizations of higher-dimensional proofs, assists in their …
This article introduces Globular, an online proof assistant for the formalization and verification of proofs in higher-dimensional category theory. The tool produces graphical visualizations of higher-dimensional proofs, assists in their construction with a point-and- click interface, and performs type checking to prevent incorrect rewrites. Hosted on the web, it has a low barrier to use, and allows hyperlinking of formalized proofs directly from research papers. It allows the formalization of proofs from logic, topology and algebra which are not formalizable by other methods, and we give several examples.
There are two main constructions in classical descent theory: the category of algebras and the descent category, which are known to be examples of weighted bilimits. We give a formal …
There are two main constructions in classical descent theory: the category of algebras and the descent category, which are known to be examples of weighted bilimits. We give a formal approach to descent theory, employing formal consequences of commuting properties of bilimits to prove classical and new theorems in the context of Janelidze-Tholen "Facets of Descent II", such as Bénabou-Roubaud Theorems, a Galois Theorem, embedding results and formal ways of getting effective descent morphisms. In order to do this, we develop the formal part of the theory on commuting bilimits via pseudomonad theory, studying idempotent pseudomonads and proving a $2$-dimensional version of the adjoint triangle theorem. Also, we work out the concept of pointwise pseudo-Kan extension, used as a framework to talk about bilimits, commutativity and the descent object. As a subproduct, this formal approach can be an alternative perspective/guiding template for the development of higher descent theory.
We define strict and weak duality involutions on 2-categories, and prove a coherence theorem that every bicategory with a weak duality involution is biequivalent to a 2-category with a strict …
We define strict and weak duality involutions on 2-categories, and prove a coherence theorem that every bicategory with a weak duality involution is biequivalent to a 2-category with a strict duality involution. For this purpose we introduce "2-categories with contravariance", a sort of enhanced 2-category with a basic notion of "contravariant morphism", which can be regarded either as generalized multicategories or as enriched categories. This enables a universal characterization of duality involutions using absolute weighted colimits, leading to a conceptual proof of the coherence theorem.
Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk's classification of regular Lie groupoids in foliation theory, …
Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk's classification of regular Lie groupoids in foliation theory, to Waldmann's work on deformation quantization. For any such action we introduce an action bicategory, together with a canonical projection (strict) 2-functor to the bicategory which acts. When the bicategory is a bigroupoid, we can impose the additional condition that action is principal in bicategorical sense, giving rise to a bigroupoid 2-torsor. In that case, the Duskin nerve of the canonical projection is precisely the Duskin-Glenn simplicial 2-torsor.
We consider a theory of centers and homotopy centers of monoids in monoidal categories which themselves are enriched in duoidal categories. Duoidal categories (introduced by Aguillar and Mahajan under the …
We consider a theory of centers and homotopy centers of monoids in monoidal categories which themselves are enriched in duoidal categories. Duoidal categories (introduced by Aguillar and Mahajan under the name 2-monoidal categories) are categories with two monoidal structures which are related by some, not necessary invertible, coherence morphisms. Centers of monoids in this sense include many examples which are not `classical.' In particular, the 2-category of categories is an example of a center in our sense. Examples of homotopy center (analogue of the classical Hochschild complex) include the Gray-category Gray of 2-categories, 2-functors and pseudonatural transformations and Tamarkin's homotopy 2-category of dg-categories, dg-functors and coherent dg-transformations.
The major improvement in this paper is that we can extend the functor $(-)^!$ of Grothendieck duality to the unbounded derived category of sufficiently nice algebraic stacks. The original motivation …
The major improvement in this paper is that we can extend the functor $(-)^!$ of Grothendieck duality to the unbounded derived category of sufficiently nice algebraic stacks. The original motivation came from formulas discovered by Avramov and Iyengar, linking Grothendieck duality with Hochscild homology and cohomology.
Given a pseudomonad $\mathcal{T} $ on a $2$-category $\mathfrak{B} $, if a right biadjoint $\mathfrak{A}\to\mathfrak{B} $ has a lifting to the pseudoalgebras $\mathfrak{A}\to\mathsf{Ps}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $ then this lifting is also right …
Given a pseudomonad $\mathcal{T} $ on a $2$-category $\mathfrak{B} $, if a right biadjoint $\mathfrak{A}\to\mathfrak{B} $ has a lifting to the pseudoalgebras $\mathfrak{A}\to\mathsf{Ps}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $ then this lifting is also right biadjoint provided that $\mathfrak{A} $ has codescent objects. In this paper, we give general results on lifting of biadjoints. As a consequence, we get a biadjoint triangle theorem which, in particular, allows us to study triangles involving the $2$-category of lax algebras, proving analogues of the result described above. In the context of lax algebras, denoting by $\ell :\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} \to\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} _\ell $ the inclusion, if $R: \mathfrak{A}\to\mathfrak{B} $ is right biadjoint and has a lifting $J: \mathfrak{A}\to \mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $, then $\ell\circ J$ is right biadjoint as well provided that $\mathfrak{A} $ has some needed weighted bicolimits. In order to prove such result, we study descent objects and lax descent objects. At the last section, we study direct consequences of our theorems in the context of the $2$-monadic approach to coherence.
By the biadjoint triangle theorem, given a pseudomonad $\mathcal{T} $ on a $2$-category $\mathfrak{B} $, if a right biadjoint $\mathfrak{A}\to\mathfrak{B} $ has a lifting to the pseudoalgebras $\mathfrak{A}\to\mathsf{Ps}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $ then …
By the biadjoint triangle theorem, given a pseudomonad $\mathcal{T} $ on a $2$-category $\mathfrak{B} $, if a right biadjoint $\mathfrak{A}\to\mathfrak{B} $ has a lifting to the pseudoalgebras $\mathfrak{A}\to\mathsf{Ps}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $ then this lifting is also right biadjoint provided that $\mathfrak{A} $ has codescent objects. In this paper, we give general results on lifting of biadjoints. As a consequence, we get a \textit{biadjoint triangle theorem} which, in particular, allows us to study triangles involving the $2$-category of lax algebras, proving analogues of the result described above. More precisely, we prove that, denoting by $\ell :\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} \to\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg}_\ell $ the inclusion, if $R: \mathfrak{A}\to\mathfrak{B} $ is right biadjoint and has a lifting $J: \mathfrak{A} \to \mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $, then $\ell\circ J$ is right biadjoint as well provided that $\mathfrak{A} $ has some needed weighted bicolimits. In order to prove such theorem, we study the descent objects and the lax descent objects. At the last section, we study direct consequences of our theorems in the context of the $2$-monadic approach to coherence. In particular, we give the construction of the left $2$-adjoint to the inclusion of the strict algebras into the lax algebras.
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.
In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. Specifically, we show that every Frobenius object in a monoidal category M arises …
In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commutative Frobenius algebra, this result also shows that every 2D TQFT is obtained from an ambijunction in some 2-category. Our theorem is proved by extending the theory of adjoint monads to the context of an arbitrary 2-category and utilizing the free completion under Eilenberg- Moore objects. We then categorify this theorem by replacing the monoidal category M with a semistrict monoidal 2-category M , and replacing the 2-category D into which it embeds by a semistrict 3-category. To state this more powerful result, we must first define the notion of a 'Frobenius pseudomonoid', which categorifies that of a Frobenius object. We then define the notion of a 'pseudo ambijunction', categorifying that of an ambijunction. In each case, the idea is that all the usual axioms now hold only up to coherent isomorphism. Finally, we show that every Frobenius pseudomonoid in a semistrict monoidal 2-category arises from a pseudo ambijunction in some semistrict 3-category.
We study the question when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an …
We study the question when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions, the category of coherent sheaves on the product of two schemes with the resolution property is given by the Deligne tensor product of the categories of coherent sheaves of the two factors. To do this we prove that the class of quasi-compact and semi-separated schemes with the resolution property is closed under fiber products.
Abstract The purpose of this survey is to present in a uniform way the notion of equivalence between strict ‐categories or ‐categories, and inside a strict ‐category or ‐category.
Abstract The purpose of this survey is to present in a uniform way the notion of equivalence between strict ‐categories or ‐categories, and inside a strict ‐category or ‐category.
Dimension three is an important test-bed for hypotheses in higher category theory and occupies something of a unique position in the categorical landscape. At the heart of matters is the …
Dimension three is an important test-bed for hypotheses in higher category theory and occupies something of a unique position in the categorical landscape. At the heart of matters is the coherence theorem, of which this book provides a definitive treatment, as well as covering related results. Along the way the author treats such material as the Gray tensor product and gives a construction of the fundamental 3-groupoid of a space. The book serves as a comprehensive introduction, covering essential material for any student of coherence and assuming only a basic understanding of higher category theory. It is also a reference point for many key concepts in the field and therefore a vital resource for researchers wishing to apply higher categories or coherence results in fields such as algebraic topology or theoretical computer science.
This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and …
This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves.
We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic string-like excitations that terminate at gapped …
We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic string-like excitations that terminate at gapped boundaries. Using a tube algebra approach, we classify such excitations and derive the corresponding representation theory. Via a dimensional reduction argument, we relate this tube algebra to that describing (2+1)d boundary point-like excitations at interfaces between two gapped boundaries. Such point-like excitations are well known to be encoded into a bicategory of module categories over the input fusion category. Exploiting this correspondence, we define a bicategory that encodes the string-like excitations ending at gapped boundaries, showing that it is a sub-bicategory of the centre of the input bicategory of group-graded 2-vector spaces. In the process, we explain how gapped boundaries in (3+1)d can be labelled by so-called pseudo-algebra objects over this input bicategory.
If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed …
If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_\otimes : \mathcal{S}\mathcal{H}(S') \to \mathcal{S}\mathcal{H}(S)$, where $\mathcal{S}\mathcal{H}(S)$ is the $\mathbb{P}^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $H\mathbb{Z}$, the homotopy $K$-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb{Z}$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.
One way of interpreting a left Kan extension is as taking a kind of "partial colimit", whereby one replaces parts of a diagram by their colimits. We make this intuition …
One way of interpreting a left Kan extension is as taking a kind of "partial colimit", whereby one replaces parts of a diagram by their colimits. We make this intuition precise by means of the partial evaluations sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the (huge) category CAT of locally small categories. Throughout, particular care is taken to handle size issues, which are notoriously delicate in the context of free cocompletion. We spell out, with all 2-dimensional details, the structure maps of these pseudomonads. Then, based on a detailed general proof of how the restriction-of-scalars construction of monads extends to the case of pseudoalgebras over pseudomonads, we consider a morphism of monads between them, which we call image. This morphism allows in particular to generalize the idea of confinal functors, i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this paper spells out how a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its center of mass.
We use the general notion of 2-dimensional adjunction with given coherence equations as introduced by MacDonald-Stone, building on earlier work by Gray, to derive coherence equations for a general 2-monad, …
We use the general notion of 2-dimensional adjunction with given coherence equations as introduced by MacDonald-Stone, building on earlier work by Gray, to derive coherence equations for a general 2-monad, which we refer to as a lax-Gray monad. The free lax-Gray 2-monad on one object may be regarded as the suspension of a lax 2-dimensional analogue of the simplicial category Delta. We call this analogue Delta LG for lax-Gray Delta. This is analogous to the way that the free 1-monad Mnd (as presented in Schanuel-Street) is a concrete example of the suspension of the simplicial category Delta, described by Mac Lane.
Kornel Szlachanyi [28] recently used the term skew-monoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized …
Kornel Szlachanyi [28] recently used the term skew-monoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skew-monoidal structures on the category of one-sided R-modules for which the lax unit was R itself. We de ne skew monoidales (or skew pseudo-monoids) in any monoidal bicategory M . These are skew-monoidal categories when M is Cat. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories [10] with base comonoid C in a suitably complete braided monoidal category V are precisely skew monoidales in Comod(V ) with unit coming from the counit of C. Quantum groupoids (in the sense of [6] rather than [10]) are those skew monoidales with invertible associativity constraint. In fact, we provide some very general results connecting opmonoidal monads and skew monoidales. We use a lax version of the concept of warping de ned in [3] to modify monoidal structures.
There is a lot of redundancy in the usual denition of adjoint functors. We dene and prove the core of what is required. First we do this in the hom-enriched …
There is a lot of redundancy in the usual denition of adjoint functors. We dene and prove the core of what is required. First we do this in the hom-enriched context. Then we do it in the cocompletion of a bicategory with respect to Kleisli objects, which we then apply to internal categories. Finally, we describe a doctrinal setting.
In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these types can be constructed from the groupoid quotient. …
In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these types can be constructed from the groupoid quotient. We define an internal notion of signatures for HITs, and for each signature, we construct a bicategory of algebras in 1-types and in groupoids. We continue by proving initial algebra semantics for our signatures. After that, we show that the groupoid quotient induces a biadjunction between the bicategories of algebras in 1-types and in groupoids. Then we construct a biinitial object in the bicategory of algebras in groupoids, which gives the desired algebra. From all this, we conclude that all finitary 1-truncated HITs can be constructed from the groupoid quotient. We present several examples of HITs which are definable using our notion of signature. In particular, we show that each signature gives rise to a HIT corresponding to the freely generated algebraic structure over it. We also start the development of universal algebra in 1-types. We show that the bicategory of algebras has PIE limits, i.e. products, inserters and equifiers, and we prove a version of the first isomorphism theorem for 1-types. Finally, we give an alternative characterization of the foundamental groups of some HITs, exploiting our construction of HITs via the groupoid quotient. All the results are formalized over the UniMath library of univalent mathematics in Coq.
Generalized operads, also called generalized multicategories and $T$-monoids, are defined as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads have been considered by numerous …
Generalized operads, also called generalized multicategories and $T$-monoids, are defined as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads have been considered by numerous authors in different contexts, with examples including symmetric multicategories, topological spaces, globular operads and Lawvere theories. In this paper we study functoriality of the Kleisli construction, and correspondingly that of generalized operads. Motivated by this problem we develop a lax version of the formal theory of monads, and study its connection to bicategorical structures.
One way of interpreting a left Kan extension is as taking a kind of "partial colimit", whereby one replaces parts of a diagram by their colimits. We make this intuition …
One way of interpreting a left Kan extension is as taking a kind of "partial colimit", whereby one replaces parts of a diagram by their colimits. We make this intuition precise by means of the "partial evaluations" sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the (huge) category CAT of locally small categories. Throughout, particular care is taken to handle size issues, which are notoriously delicate in the context of free cocompletion. We spell out, with all 2-dimensional details, the structure maps of these pseudomonads. Then, based on a detailed general proof of how the "restriction-of-scalars" construction of monads extends to the case of pseudoalgebras over pseudomonads, we define a morphism of monads between them, which we call "image". This morphism allows us in particular to generalize the idea of "confinal functors" i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this paper spells out how a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its center of mass.
We introduce the notion of a relative pseudomonad, which generalizes the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient …
We introduce the notion of a relative pseudomonad, which generalizes the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonads on the Kleisli bicategory of a relative pseudomonad. The results are applied to define several pseudomonads on the bicategory of profunctors in an homogeneous way and provide a uniform approach to the definition of bicategories that are of interest in operad theory, mathematical logic, and theoretical computer science.
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.
Abstract In 2009, Lipman published his polished book (Lipman in Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Mathematics, vol. 1960, Springer, Berlin, pp 1–259, 2009), the …
Abstract In 2009, Lipman published his polished book (Lipman in Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Mathematics, vol. 1960, Springer, Berlin, pp 1–259, 2009), the product of a decade’s work, giving the definitive, state-of-the-art treatment of Grothendieck duality. In this article, we achieve a sharp improvement: we begin by giving a new proof of the base-change theorem, which can handle unbounded complexes and work in the generality of algebraic stacks (subject to mild technical restrictions). This means that our base-change theorem must be subtle and delicate, the unbounded version is right at the boundary of known counterexamples—counterexamples (in the world of schemes) that had led the experts to believe that major parts of the theory could only be developed in the bounded-below derived category. Having proved our new base-change theorem, we then use it to define the functor $$f^!$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>f</mml:mi> <mml:mo>!</mml:mo> </mml:msup> </mml:math> on the unbounded derived category and establish its functoriality properties. In Sect. 1, we will use this to clarify the relation among all the various constructions of Grothendieck duality. One illustration of the power of the new methods is that we can improve Lipman (Lecture Notes in Mathematics, vol. 1960, Springer, Berlin, pp 1–259, 2009, Theorem 4.9.4) to handle complexes that are not necessarily bounded. There are also applications to the theory developed by Avramov, Iyengar, Lipman and Nayak on the connection between Grothendieck duality and Hochschild homology and cohomology but, to keep this paper from becoming even longer, these are being relegated to separate articles. See for example (Neeman in K -Theory—Proceedings of the International Colloquium, Mumbai, 2016, Hindustan Book Agency, New Delhi, pp 91–126, 2018).
We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples …
We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of univalent bicategories in a modular fashion, we develop displayed bicategories, an analog of displayed 1-categories introduced by Ahrens and Lumsdaine. We demonstrate the applicability of this notion, and prove that several bicategories of interest are univalent. Among these are the bicategory of univalent categories with families and the bicategory of pseudofunctors between univalent bicategories. Furthermore, we show that every bicategory with univalent hom-categories is weakly equivalent to a univalent bicategory. All of our work is formalized in Coq as part of the UniMath library of univalent mathematics.
We pursue the definition of a KZ-doctrine in terms of a fully faithful adjoint string Dd ⊣ m ⊣ dD. We give the definition in any Gray-category. The concept of …
We pursue the definition of a KZ-doctrine in terms of a fully faithful adjoint string Dd ⊣ m ⊣ dD. We give the definition in any Gray-category. The concept of algebra is given as an adjunction with invertible counit. We show that these doctrines are instances of more general pseudomonads. The algebras for a pseudomonad are defined in more familiar terms and shown to be the same as the ones defined as adjunctions when we start with a KZ-doctrine.
Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20.
Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20.