Linear bicategories are a generalization of bicategories in which the one horizontal composition is replaced by two (linked) horizontal compositions. These compositions provide a semantic model for the tensor and …
Linear bicategories are a generalization of bicategories in which the one horizontal composition is replaced by two (linked) horizontal compositions. These compositions provide a semantic model for the tensor and par of linear logic: in particular, as composition is fundamentally non-commutative, they provide a suggestive source of models for non-commutative linear logic. In a linear bicategory, the logical notion of complementation becomes a natural linear notion of adjunction. Just as ordinary adjoints are related to (Kan) extensions, these linear adjoints are related to the appropriate notion of linear extension. There is also a stronger notion of complementation, which arises, for example, in cyclic linear logic. This sort of complementation is modelled by cyclic adjoints. This leads to the notion of a *ast;-linear bicategory and the coherence conditions that it must satisfy. Cyclic adjoints also give rise to linear monads: these are, essentially, the appropriate generalization (to the linear setting) of Frobenius algebras and the ambialgebras of Topological Quantum Field Theory. A number of examples of linear bicategories arising from different sources are described, and a number of constructions that result in linear bicategories are indicated.
A concise guide to very basic bicategory theory, from the definition of a bicategory to the coherence theorem.
A concise guide to very basic bicategory theory, from the definition of a bicategory to the coherence theorem.
A concise guide to very basic bicategory theory, from the definition of a bicategory to the coherence theorem.
A concise guide to very basic bicategory theory, from the definition of a bicategory to the coherence theorem.
It is well known that to give an oplax functor of bicategories $\mathbf{1}\to\mathscr{C}$ is to give a comonad in $\mathscr{C}$. Here we generalize this fact, replacing the terminal bicategory by …
It is well known that to give an oplax functor of bicategories $\mathbf{1}\to\mathscr{C}$ is to give a comonad in $\mathscr{C}$. Here we generalize this fact, replacing the terminal bicategory by any bicategory $\mathscr{A}$ for which the composition functor admits generic factorisations. We call bicategories with this property generic, and show that for generic bicategories $\mathscr{A}$ one may express the data of an oplax functor $\mathscr{A}\to\mathscr{C}$ much like the data of a comonad; the main advantage of this description being that it does not directly involve composition in $\mathscr{A}$. We then go on to apply this result to some well known bicategories, such as cartesian monoidal categories (seen as one object bicategories), bicategories of spans, and bicategories of polynomials with cartesian 2-cells.
It is well known that to give an oplax functor of bicategories $\mathbf{1}\to\mathscr{C}$ is to give a comonad in $\mathscr{C}$. Here we generalize this fact, replacing the terminal bicategory by …
It is well known that to give an oplax functor of bicategories $\mathbf{1}\to\mathscr{C}$ is to give a comonad in $\mathscr{C}$. Here we generalize this fact, replacing the terminal bicategory by any bicategory $\mathscr{A}$ for which the composition functor admits generic factorisations. We call bicategories with this property generic, and show that for generic bicategories $\mathscr{A}$ one may express the data of an oplax functor $\mathscr{A}\to\mathscr{C}$ much like the data of a comonad; the main advantage of this description being that it does not directly involve composition in $\mathscr{A}$. We then go on to apply this result to some well known bicategories, such as cartesian monoidal categories (seen as one object bicategories), bicategories of spans, and bicategories of polynomials with cartesian 2-cells.
Abstract In this chapter, the tricategory of bicategories is presented in full detail. After a preliminary discussion of the whiskerings of a lax transformation with a lax functor, the chapter …
Abstract In this chapter, the tricategory of bicategories is presented in full detail. After a preliminary discussion of the whiskerings of a lax transformation with a lax functor, the chapter goes on to define a tricategory. The rest of the chapter proves in detail the existence of a tricategory with small bicategories as objects (i.e. a tricategory of bicategories), pseudofunctors as 1-cells, strong transformations as 2-cells, and modifications as 3-cells.
In a Brown category of cofibrant objects, there is a model for the mapping spaces of the hammock localization in terms of zig-zags of length 2. In this paper we …
In a Brown category of cofibrant objects, there is a model for the mapping spaces of the hammock localization in terms of zig-zags of length 2. In this paper we show how to assemble these spaces into a Segal category that models the infinity-categorical localization of the Brown category.
In a Brown category of cofibrant objects, there is a model for the mapping spaces of the hammock localization in terms of zig-zags of length 2. In this paper we …
In a Brown category of cofibrant objects, there is a model for the mapping spaces of the hammock localization in terms of zig-zags of length 2. In this paper we show how to assemble these spaces into a Segal category that models the infinity-categorical localization of the Brown category.
In a Brown category of cofibrant objects, there is a model for the mapping spaces of the hammock localization in terms of zig-zags of length 2. In this paper we …
In a Brown category of cofibrant objects, there is a model for the mapping spaces of the hammock localization in terms of zig-zags of length 2. In this paper we show how to assemble these spaces into a Segal category that models the infinity-categorical localization of the Brown category.
Linearly distributive categories were introduced to model the tensor/par fragment of linear logic, without resorting to the use of negation. Linear bicategories are the bicategorical version of linearly distributive categories. …
Linearly distributive categories were introduced to model the tensor/par fragment of linear logic, without resorting to the use of negation. Linear bicategories are the bicategorical version of linearly distributive categories. Essentially, a linear bicategory has two forms of composition, each determining the structure of a bicategory, and the two compositions are related by a linear distribution. While it is standard in the field of monoidal topology that the category of quantale-valued relations is a bicategory, we begin by showing that if the quantale is a Girard quantale, we obtain a linear bicategory. We further show that the category $QRel$ for $Q$ a unital quantale is a Girard quantaloid if and only if $Q$ is a Girard quantale. The tropical and arctic semiring structures fit together into a Girard quantale, so this construction is likely to have multiple applications. More generally, we define LD-quantales, which are sup-lattices with two quantale structures related by a linear distribution, and show that $QRel$ is a linear bicategory if $Q$ is an LD-quantale. We then consider several standard constructions from bicategory theory, and show that these lift to the linear bicategory setting and produce new examples of linear bicategories. In particular, we consider quantaloids. We first define the notion of a linear quantaloid ${\cal Q}$ and then consider linear ${\cal Q}$-categories and linear monads in ${\cal Q}$, where ${\cal Q}$ is a linear quantaloid. Every linear quantaloid is a linear bicategory. We also consider the bicategory whose objects are locales, 1-cells are bimodules and two-cells are bimodule homomorphisms. This bicategory turns out to be what we call a Girard bicategory, which are in essence a closed version of linear bicategories.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
We define bicategories internal to 2-categories. When the ambient 2-category is symmetric monoidal categories, this provides a convenient framework for encoding the structures of a symmetric monoidal 3-category. This framework …
We define bicategories internal to 2-categories. When the ambient 2-category is symmetric monoidal categories, this provides a convenient framework for encoding the structures of a symmetric monoidal 3-category. This framework is well suited to examples arising in geometry and algebra, such as the 3-category of bordisms or the 3-category of conformal nets.
In this paper, we establish some connections between the concept of an equivalence of categories and that of an equivalence in a bicategory. Its main result builds upon the observation …
In this paper, we establish some connections between the concept of an equivalence of categories and that of an equivalence in a bicategory. Its main result builds upon the observation that two closely related concepts, which could both play the role of an equivalence in a bicategory, turn out not to coincide. Two counterexamples are provided for that goal, and detailed proofs are given. In particular, all calculations done in a bicategory are fully explicit, in order to overcome the difficulties which arise when working with bicategories instead of 2-categories.
Abstract In this chapter, the Whitehead Theorem for bicategories is proved in detail. The Whitehead Theorem states that a pseudofunctor between bicategories is a biequivalence if and only if it …
Abstract In this chapter, the Whitehead Theorem for bicategories is proved in detail. The Whitehead Theorem states that a pseudofunctor between bicategories is a biequivalence if and only if it is surjective up to adjoint equivalences on objects, surjective up to isomorphisms on 1-cells, and bijective on 2-cells. The chapter covers the lax slice bicategory, lax terminal objects, and the Quillen Theorem A for bicategories. A 2-categorical version of the Whitehead Theorem is also discussed.
Abstract In this chapter, 2-categories and bicategories are defined, along with basic examples. Several useful unity properties in bicategories, generalizing those in monoidal categories and underlying many fundamental results in …
Abstract In this chapter, 2-categories and bicategories are defined, along with basic examples. Several useful unity properties in bicategories, generalizing those in monoidal categories and underlying many fundamental results in bicategory theory, are discussed. In addition to well-known examples, the 2-categories of multicategories and of polycategories are constructed. This chapter ends with a discussion of duality of bicategories.
We study the existence and uniqueness of direct sum decompositions in additive bicategories. We find a simple definition of Krull-Schmidt bicategories, for which we prove the uniqueness of decompositions into …
We study the existence and uniqueness of direct sum decompositions in additive bicategories. We find a simple definition of Krull-Schmidt bicategories, for which we prove the uniqueness of decompositions into indecomposable objects as well as a characterization in terms of splitting of idempotents and properties of 2-cell endomorphism rings. Examples of Krull-Schmidt bicategories abound, with many arising from the various flavors of 2-dimensional linear representation theory.
We now discuss a class of results related to universal constructions which we call the Hoffman-Newman-Radford (HNR) rigidity theorems. They come in different flavors which we explain one by one. …
We now discuss a class of results related to universal constructions which we call the Hoffman-Newman-Radford (HNR) rigidity theorems. They come in different flavors which we explain one by one. In each case, the result provides explicit inverse isomorphisms between two universally constructed bimonoids. We call these the Hoffman-Newman-Radford (HNR) isomorphisms. For a cocommutative comonoid, the free bimonoid on that comonoid is isomorphic to the free bimonoid on the same comonoid but with the trivial coproduct. The product is concatenation in both, but the coproducts differ, it is dequasishuffle in the former and deshuffle in the latter. An explicit isomorphism can be constructed in either direction, one direction involves a noncommutative zeta function, while the other direction involves a noncommutative Möbius function.These are the HNR isomorphisms. There is a dual result starting with a commutative monoid.In this case, the coproduct is deconcatenation in both, but the products differ, it is quasishuffle in the former and shuffle in the latter. Interestingly, these ideas can be used to prove that noncommutative zeta functions and noncommutative Möbius functions are inverse to each other in the lune-incidence algebra. There is a commutative analogue of the above results in which the universally constructed bimonoids are bicommutative. Now the HNR isomorphisms are constructed using the zeta function and Möbius function of the poset of flats. As an application, we explain how they can be used to diagonalize the mixed distributive law for bicommutative bimonoids. There is also a q-analogue, for q not a root of unity. In this case, the HNR isomorphisms involve the two-sided q-zeta and q-Möbius functions. As an application, we explain how they can be used to study the nondegeneracy of the mixed distributive law for q-bimonoids.
Abstract This is the third paper of this series. In Wang [Monopoles and Landau‐Ginzburg models II: Floer homology. arXiv:2005.04333, 2020], we defined the monopole Floer homology for any pair , …
Abstract This is the third paper of this series. In Wang [Monopoles and Landau‐Ginzburg models II: Floer homology. arXiv:2005.04333, 2020], we defined the monopole Floer homology for any pair , where is a compact oriented 3‐manifold with toroidal boundary and is a suitable closed 2‐form viewed as a decoration. In this paper, we establish a gluing theorem for this Floer homology when two such 3‐manifolds are glued suitably along their common boundary, assuming that is disconnected, and is small and yet non‐vanishing on . As applications, we construct a monopole Floer 2‐functor and the generalized cobordism maps. Using results of Kronheimer–Mrowka and Ni, it is shown that for any such 3‐manifold that is irreducible, this Floer homology detects the Thurston norm on and the fiberness of . Finally, we show that our construction recovers the monopole link Floer homology for any link inside a closed 3‐manifold.
Corings and comodules were defined by Sweedler in 1975 and are defined as coalgebras (or co-monoids) in the monoidal category of bimodules over a (non-commutative) ring. In this way they …
Corings and comodules were defined by Sweedler in 1975 and are defined as coalgebras (or co-monoids) in the monoidal category of bimodules over a (non-commutative) ring. In this way they are, from a categorical point of view, the dual notion of a ring (extension). The most interesting examples of corings were found only in 1999 by Takeuchi, who observed that corings and comodules can be constructed out of entwining structures and entwined modules, and therefore as well out of all kinds of structures that were intensively studied previously in Hopf algebra theory. In fact, by passing from entwining structures and entwined modules to corings and comodules, it turned out that most structural theorems could be preserved, moreover calculations became easier and more transparent and many results could be clarified as they were presented in a broader perspective. Galois theory for corings and comodules orriginates from the classical work on Galois extensions of commutative fields. A group action can be generalized to a Hopf algebra (co)action, which has lead to the development of Hopf-Galois theory. Hopf-Galois theory as well has a formulation in terms of corings and is one of the major research subjects in present coring theory. It allows one to characterize in general equivalences between categories of comodules over a coring and modules over a (possibly non-unital firm) ring.
Abstract The aim of the paper is to introduce an approach to the theory of 2-categories which is based on systematic use of the Grothendieck construction and the Segal Machine …
Abstract The aim of the paper is to introduce an approach to the theory of 2-categories which is based on systematic use of the Grothendieck construction and the Segal Machine and to show how adjunction questions can be investigated by means of this approach and what its connections are with more traditional approaches. As an application, the derived Morita 2-category and the Fourier–Mukai 2-category over a Noetherian ring are constructed and the embedding of the latter in the former is demonstrated. Bibliography: 15 titles.
Given a finitely generated and projective Lie–Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its …
Given a finitely generated and projective Lie–Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the appendices we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings.
We show that two flat commutative Hopf algebroids are Morita equivalent if and only if they are weakly equivalent and if and only if there exists a principal bibundle connecting …
We show that two flat commutative Hopf algebroids are Morita equivalent if and only if they are weakly equivalent and if and only if there exists a principal bibundle connecting them. This gives a positive answer to a conjecture due to Hovey and Strickland. We also prove that principal (left) bundles lead to a bicategory together with a 2-functor from flat Hopf algebroids to trivial principal bundles. This turns out to be the universal solution for 2-functors which send weak equivalences to invertible 1-cells. Our approach can be seen as an algebraic counterpart to Lie groupoid Morita theory.
We give a formal concept of (right) wide Morita context between two 0-cells in arbitrary bicategory. We then construct a new bicategory with the same 0-cells as the oldest one, …
We give a formal concept of (right) wide Morita context between two 0-cells in arbitrary bicategory. We then construct a new bicategory with the same 0-cells as the oldest one, and with 1-cells all these (right) wide Morita contexts. An application to the (right) Eilenberg-Moore bicategory of comonads associated to the bimodules bicategory is also given.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
This chapter provides a categorical framework for the notions of monoids, comonoids, bimonoids in species (relative to a fixed hyperplane arrangement). The usual categorical setting for monoids is a monoidal …
This chapter provides a categorical framework for the notions of monoids, comonoids, bimonoids in species (relative to a fixed hyperplane arrangement). The usual categorical setting for monoids is a monoidal category. However, that is not the case here; the relevant concept is that of monads and algebras over monads. We construct a monad on the category of species, and observe that algebras over it are the same as monoids in species. Dually, we construct a comonad whose coalgebras are the same as comonoids in species. In addition, we construct a mixed distributive law between this monad and comonad such that bialgebras over the resulting bimonad are the same as bimonoids in species. Moreover, the mixed distributive law can be deformed by a parameter q such that the resulting bialgebras are the same as q-bimonoids. The above monad, comonad, bimonad have commutative counterparts which relate to commutative monoids, cocommutative comonoids, bicommutative bimonoids in species. We briefly discuss the Mesablishvili-Wisbauer rigidity theorem. As a consequence, the category of species is equivalent to the category of 0-bimonoids, as well as to the category of bicommutative bimonoids. These ideas are developed in more detail later. We extend the notion of species from a hyperplane arrangement to the more general setting of a left regular band (LRB).
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
We extend the theory of Sweeder's measuring comonoids to the framework of duoidal categories: categories equipped with two compatible monoidal structures. We use one of the tensor products to endow …
We extend the theory of Sweeder's measuring comonoids to the framework of duoidal categories: categories equipped with two compatible monoidal structures. We use one of the tensor products to endow the category of monoids for the other with an enrichment in the category of comonoids. The enriched homs are provided by the universal measuring comonoids. We study a number of duoidal structures on categories of graded objects and of species and the associated enriched categories, such as an enrichment of graded (twisted) monoids in graded (twisted) comonoids, as well as two enrichments of symmetric operads in symmetric cooperads.
This is the third paper of this series. In \cite{Wang20}, we defined the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary …
This is the third paper of this series. In \cite{Wang20}, we defined the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ is a suitable closed 2-form viewed as a decoration. In this paper, we establish a gluing theorem for this Floer homology when two such 3-manifolds are glued suitably along their common boundary, assuming that $\partial Y$ is disconnected, and $\omega$ is small and yet non-vanishing on $\partial Y$. As applications, we construct a monopole Floer 2-functor and the generalized cobordism maps. Using results of Kronheimer-Mrowka and Ni, it is shown that for any such 3-manifold $Y$ that is irreducible, this Floer homology detects the Thurston norm on $H_2(Y,\partial Y;\mathbb{R})$ and the fiberness of $Y$. Finally, we show that our construction recovers the monopole link Floer homology for any link inside a closed 3-manifold.
We define weak units in a semi-monoidal 2-category \mathcal C as cancellable pseudo-idempotents: they are pairs (I,\alpha) where I is an object such that tensoring with I from either side …
We define weak units in a semi-monoidal 2-category \mathcal C as cancellable pseudo-idempotents: they are pairs (I,\alpha) where I is an object such that tensoring with I from either side constitutes a biequivalence of \mathcal C , and \alpha: I \otimes I \to I is an equivalence in \mathcal C . We show that this notion of weak unit has coherence built in: Theorem A: \alpha has a canonical associator 2-cell, which automatically satisfies the pentagon equation. Theorem B: every morphism of weak units is automatically compatible with those associators. Theorem C: the 2-category of weak units is contractible if non-empty. Finally we show (Theorem E) that the notion of weak unit is equivalent to the notion obtained from the definition of tricategory: \alpha alone induces the whole family of left and right maps (indexed by the objects), as well as the whole family of Kelly 2-cells (one for each pair of objects), satisfying the relevant coherence axioms.
In this paper we give an elementary derivation of a 2-groupoid from a fibration.This extends a previous result for pointed fibrations due to Loday.Discussion is included as to the translation …
In this paper we give an elementary derivation of a 2-groupoid from a fibration.This extends a previous result for pointed fibrations due to Loday.Discussion is included as to the translation between 2-groupoids and cat 1 -groupoids.where
For any small quantaloid $\Q$, there is a new quantaloid $\D(\Q)$ of diagonals in $\Q$. If $\Q$ is divisible then so is $\D(\Q)$ (and vice versa), and then it is …
For any small quantaloid $\Q$, there is a new quantaloid $\D(\Q)$ of diagonals in $\Q$. If $\Q$ is divisible then so is $\D(\Q)$ (and vice versa), and then it is particularly interesting to compare categories enriched in $\Q$ with categories enriched in $\D(\Q)$. Taking Lawvere's quantale of extended positive real numbers as base quantale, $\Q$-categories are generalised metric spaces, and $\D(\Q)$-categories are generalised partial metric spaces, i.e.\ metric spaces in which self-distance need not be zero and with a suitably modified triangular inequality. We show how every small quantaloid-enriched category has a canonical closure operator on its set of objects: this makes for a functor from quantaloid-enriched categories to closure spaces. Under mild necessary-and-sufficient conditions on the base quantaloid, this functor lands in the category of topological spaces; and an involutive quantaloid is Cauchy-bilateral (a property discovered earlier in the context of distributive laws) if and only if the closure on any enriched category is identical to the closure on its symmetrisation. As this now applies to metric spaces and partial metric spaces alike, we demonstrate how these general categorical constructions produce the "correct" definitions of convergence and Cauchyness of sequences in generalised partial metric spaces. Finally we describe the Cauchy-completion, the Hausdorff contruction and exponentiability of a partial metric space, again by application of general quantaloid-enriched category theory.
Abstract In this study, we will express the 2-crossed module of groups from a higher-dimensional categorical perspective. According to simplicial homotopy theory, a 2-crossed module is the Moore complex of …
Abstract In this study, we will express the 2-crossed module of groups from a higher-dimensional categorical perspective. According to simplicial homotopy theory, a 2-crossed module is the Moore complex of a 2-truncated simplicial group. Therefore, the 2-crossed module is an algebraic homotopy model for the homotopy 3-types. Tricategories are a three-dimensional generalization of the bicategory concept. Any tricategory is triequivalent to the Gray category, where Gray is a category enriched over the monoidal category 2Cat equipped with the Gray tensor product. Briefly, a Gray category is a semi-strict 3-category for homotopy 3-types. Naturally, the tricategory perspective is used in homotopy theory. The 2-crossed module is associated with the concept of the Gray category. The aim of this study is to obtain a single object tricategory from any 2-crossed module of groups.
A riple F (F, , ) in ctegory a consists of functor F a nd morphisms la F, F F stisfying some identities (see 2, (T.1)- (T.3)) nlogous to those …
A riple F (F, , ) in ctegory a consists of functor F a nd morphisms la F, F F stisfying some identities (see 2, (T.1)- (T.3)) nlogous to those stisfied in monoid.Cotriples re defined dually.It has been recognized by Huber [4] that whenever one hs pir of adoint functors T a , S a (see 1), then the functor TS (with appro- priate morphisms resulting from the adjointness relation) constitutes a triple in nd similarly ST yields cotriple in a.The main objective of this pper is to show that this relation between d- jointness nd triples is in some sense reversible.Given triple Y in a we de- fine new ctegory a nd adoint functors T a a, S a a such that the triple given by TS coincides with .There my be mny adoint pirs which in this wy generate the triple Y, but among those there is a uni- versal one (which therefore is in a sense the "best possible one") nd for this one the functor T is faithful (Theorem 2.2).This construction cn best be illustrated by n example.Let a be the ctegory of modules over a commu- tative ring K nd let A be K-lgebm.The functor F A@ together with morphisms nd resulting from the morphisms K A, h @ A A given by the K-algebra structure of A, yield then a triple Y a.The ctegory a is then precisely the ctegory of A-modules.The general construction of a closely resembles this example.As another example, let a be the category of sets nd let F be the functor which to ech set A ssigns the underlying set of the free group generated by A. There results triple Y in a nd a is the category of groups. Let G(, e, G) be cotriple in category A. It has been recognized by Godement [3] and Huber [4], that the iterates G of G together with face and degeneracy morphisms G + G , G G + defined using e and yield a simplicial structure which can be used to define homology and cohomology.Now if Y is a triple in a, then one has an adjoint pair T" aa, S and therefore one has an associated cotriple G in .This in turn yields a simplicial complex for every object in a , thus paving the way for homology and cohomology in ar.In 4 we show that under suitable