Closed monoidal category

Summary

In mathematics, especially in , a closed monoidal category (or a monoidal closed category) is a that is both a and a in such a way that the structures are compatible. A classic example is the , Set, where the monoidal product of sets and is the usual cartesian product , and the internal Hom is the set of functions from to . A non- example is the , K-Vect, over a field . Here the monoidal product is the usual tensor product of vector spaces, and the internal Hom is the vector space of linear maps from one vector space to another. The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system. Many examples of closed monoidal categories are . However, this need not always be the case, as non-symmetric monoidal categories can be encountered in category-theoretic formulations of linguistics; roughly speaking, this is because word-order in natural language matters. A closed monoidal category is a such that for every object the functor given by right tensoring with has a right adjoint, written This means that there exists a bijection, called 'currying', between the Hom-sets that is natural in both A and C. In a different, but common notation, one would say that the functor has a right adjoint Equivalently, a closed monoidal category is a category equipped, for every two objects A and B, with an object , a morphism , satisfying the following universal property: for every morphism there exists a unique morphism such that It can be shown that this construction defines a functor . This functor is called the internal Hom functor, and the object is called the internal Hom of and . Many other notations are in common use for the internal Hom. When the tensor product on is the cartesian product, the usual notation is and this object is called the exponential object. Strictly speaking, we have defined a right closed monoidal category, since we required that right tensoring with any object has a right adjoint.

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https://en.wikipedia.org/wiki/Closed_monoidal_category

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Homotopic Descent over Monoidal Model Categories

The starting point for this project is the article of Kathryn Hess [11]. In this article, a homotopic version of monadic descent is developed. In the classical setting, one constructs a category D(𝕋) of coalgebras in the Eilenberg-Moore category of algebras D𝕋 for a given monad 𝕋 on a category D. There is a canonical functor Can𝕃𝕋 from D to D(𝕋), and if Can𝕃𝕋 is fully faithful, then 𝕋 satisfies descent, while if Can𝕃𝕋 is an equivalence of categories, then 𝕋 satisfies effective descent [19]. In [11], these two conditions are replaced by a weaker one, that these hold only up to homotopy. This is achieved by working with model categories that are enriched over simplicial sets. Homotopic descent is then defined by demanding that each component in (Can𝕃𝕋)A,B : MapD(A,B) → MapD(𝕋) (Can𝕃𝕋(A), Can𝕃𝕋 (B)) be a weak equivalence of simplicial sets. A similar but stronger condition involving the path components in D(𝕋) expresses effective homotopic descent. The first goal of this project is to develop a framework of homotopic descent for model categories that are enriched over model categories other than simplicial sets. The most important examples we have in mind are chain complexes and spectra. In order to achieve this goal, we tried to determine the most general conditions that are sufficient and necessary to make the theory work. To ease the formulation, let us say that we are working with a model category D that is enriched over a monoidal model category V. The crucial constructions we need are realization, respectively totalization, of (co)simplicial objects in D. These functors have to be Quillen functors to ensure that they have the correct homotopical behaviour. This implies that there must exist a Quillen adjunction between V and simplicial sets. Furthermore, we need to be able to transfer the enrichment and (co)tensoring over V to an enrichment and (co)tensoring over simplicial sets. This forces the Quillen adjunction to be monoidal. Another main point that has to be adressed is the question, of whether the enrichment of D carries over to an enrichment of D𝕋 and D(𝕋) and how this enrichment behaves. It turns out that this works well under mild assumptions on V. This leads then to the definition of homotopic descent by requiring that each component in (Can𝕃𝕋)A,B : MapD(A,B) → MapD(𝕋) (Can𝕃𝕋(A), Can𝕃𝕋 (B)) be a weak equivalence in M and similarly for effective homotopic descent. Using this definition, the theorems in [11] carry over to this more general context. Although the conditions on V are rather constraining regarding the relation with simplicial sets, the cases of chain complexes and spectra are included. For the time being we do not see how the constraints on V could be weakened. The second goal of this project is to apply the theory of homotopic descent to concrete examples. A good source of examples is homotopic Grothendieck descent in the category of spectra, i.e., S-modules. Classical Grothendieck descent deals with the adjunction induced by a morphism φ : B → A of monoids in a monoidal category (M,Λ, S), – BΛ A : ModB ⇄ ModA : φ*, which in turn induces a monad 𝕋φ := φ*(– ΛB A) on ModB. We consider in particular the case when the morphism in question is the unit of an S-algebra E, η : S → E There is a close relationship between comodules over a Hopf algebroid and objects in D(𝕋η). Associated to η we have the canonical co-ring Wη := E ΛS E and an isomorphism between D(𝕋η) and the category of comodules over Wη in the category of S-modules. This relationship is explored in an analysis of the stable Adams spectral sequence, the construction of which heavily relies on the monadic properties of the functor η*(E ΛS –) and can therefore be expressed in terms of D(𝕋η). We construct a spectral sequence that generalizes the stable Adams spectral sequence to any stable pointed model category such as unbounded chain complexes. One can give a description of the E2-term as an Ext in D(𝕋η), E2s,t = ExtD(𝕋η) (Can(A), Can(B)). If the spectral sequences converges, it abuts to π⁎MapD(A,B η^), where Bη^ is the derived 𝕋η-completion of B, which agrees with the usual derived completion in well-known special cases. Furthermore, Bη^ := Tot B^•, and B^• is kind of a fibrant cosimplicial resolution of B. Furthermore, the language of relative homological algebra for modules and comodules generalizes to definitions for algebras in D𝕋η and coalgebras in D(𝕋η). This shows that the construction of the Adams spectral sequence works in a more general setting, where one applies a functor to an abelian category, for example π⁎, only at the end, to be able to do computations in homological algebra. This general Adams spectral sequence is closely related to the descent spectral sequence of [11], and we have clarified this relationship.

EPFL2011

K-Theory was originally defined by Grothendieck as a contravariant functor from a subcategory of schemes to abelian groups, known today as K0. The same kind of construction was then applied to other fields of mathematics, like spaces and (not necessarily commutative) rings. In all these cases, it consists of some process applied, not directly to the object one wants to study, but to some category related to it: the category of vector bundles over a space, of finitely generated projective modules over a ring, of locally free modules over a scheme, for instance. Later, Quillen extracted axioms that all these categories satisfy and that allow the Grothendieck construction of K0. The categorical structure he discovered is called today a Quillen-exact category. It led him not only to broaden the domain of application of K-theory, but also to define a whole K-theory spectrum associated to such a category. Waldhausen next generalized Quillen's notion of an exact category by introducing categories with weak equivalences and cofibrations, which one nowadays calls Waldhausen categories. K-theory has since been studied as a functor from the category of suitably structured (Quillen-exact, Waldhausen, symmetric monoidal) small categories to some category of spectra1. This has given rise to a huge field of research, so much so that there is a whole journal devoted to the subject. In this thesis, we want to take advantage of these tools to begin studying K-theory from another perspective. Indeed, we have the impression that, in the generalization of topological and algebraic K-theory that has been started by Quillen, something important has been left aside. K-theory was initiated as a (contravariant) functor from the various categories of spaces, rings, schemes, …, not from the category of Waldhausen small categories. Of course, one obtains information about a ring by studying its Quillen-exact category of (finitely generated projective) modules, but still, the final goal is the study of the ring, and, more globally, of the category of rings. Thus, in a general theory, one should describe a way to associate not only a spectrum to a structured category, but also a structured category to an object. Moreover, this process should take the morphisms of these objects into account. This gives rise to two fundamental questions. What kind of mathematical objects should K-theory be applied to? Given such an object, what category "over it" should one consider and how does it vary over morphisms? Considering examples, we have made the following observations. Suppose C is the category that is to be investigated by means of K-theory, like the category of topological spaces or of schemes, for instance. The category associated to an object of C is a sub-category of the category of modules over some monoid in a monoidal category with additional structure (topological, symmetric, abelian, model). The situation is highly "fibred": not only morphisms of C induce (structured) functors between these sub-categories of modules, but the monoidal category in which theses modules take place might vary from one object of C to another. In important cases, the sub-categories of modules considered are full sub-categories of "locally trivial" modules with respect to some (possibly weakened notion of) Grothendieck topology on C . That is, there are some specific modules that are considered sufficiently simple to be called trivial and locally trivial modules are those that are, locally over a covering of the Grothendieck topology, isomorphic to these. In this thesis, we explore, with K-theory in view, a categorical framework that encodes these kind of data. We also study these structures for their own sake, and give examples in other fields. We do not mention in this abstract set-theoretical issues, but they are handled with care in the discussion. Moreover, an appendix is devoted to the subject. After recalling classical facts of Grothendieck fibrations (and their associated indexed categories), we provide new insights into the concept of a bifibration. We prove that there is a 2-equivalence between the 2-category of bifibrations over a category ℬ and a 2-category of pseudo double functors from ℬ into the double category of adjunctions in CAT. We next turn our attention to composable pairs of fibrations , as they happen to be fundamental objects of the theory. We give a characterization of these objects in terms of pseudo-functors ℬop → FIBc into the 2-category of fibrations and Cartesian functors. We next turn to a short survey about Grothendieck (pre-)topologies. We start with the basic notion of covering function, that associate to each object of a category a family of coverings of the object. We study separately the saturation of a covering function with respect to sieves and to refinements. The Grothendieck topology generated by a pretopology is shown to be the result of these two steps. We define then, inspired by Street [89], the notion of (locally) trivial objects in a fibred category P : ℰ → ℬ equipped with some notion of covering of objects of the base ℬ. The trivial objects are objects chosen in some fibres. An object E in the fibre over B ∈ ℬ is locally trivial if there exists a covering {fi : Bi → B}i ∈ I such the inverse image of E along fi is isomorphic to a trivial object. Among examples are torsors, principal bundles, vector bundles, schemes, locally constant sheaves, quasi-coherent and locally free sheaves of modules, finitely generated projective modules over commutative rings, topological manifolds, … We give conditions under which locally trivial objects form a subfibration of P and describe the relationship between locally trivial objects with respect to subordinated covering functions. We then go into the algebraic part of the theory. We give a definition of monoidal fibred categories and show a 2-equivalence with monoidal indexed categories. We develop algebra (monoids and modules) in these two settings. Modules and monoids in a monoidal fibred category ℰ → ℬ happen to form a pair of fibrations . We end this thesis by explaining how to apply this categorical framework to K-theory and by proposing some prospects of research. ______________________________ 1 Works of Lurie, Toën and Vezzosi have shown that K-theory really depends on the (∞, 1)-category associated to a Waldhausen category [94]. Moreover, topological K-theory of spaces and Banach algebras takes the fact that the Waldhausen category is topological in account [62, 70].

EPFL2010

Coalgèbres d'Alexander-Whitney

Théophile Naïto

The goal of this work is to study Alexander-Whitney coalgebras (first defined in [HPST06]) from a topological point of view. An Alexander-Whitney coalgebra is a coassociative chain coalgebra over Z with an extra algebraic structure : the comultiplication must respect the coalgebra structure up to an infinite sequence of homotopies (this sequence is part of the data of the Alexander-Whitney coalgebra structure). Alexander-Whitney coalgebras are interesting for topologists because the normalized chain complex C(K) of a simplicial set K is endowed with an Alexander-Whitney coalgebra structure. This theorem is proved for the first time here (generalising a result proven in [HPST06]). This theorem gives the hope that the Alexander-Whitney coalgebra structure of C(K) contains interesting information that can be used to solve topological problems. This hope is strengthened by the success already obtained in the work of several topologists. Among others, [HPST06], [HL07], [Boy08], and [HR] use the Alexander-Whitney coalgebra structure of the normalized chains of a simplicial set in an essential way to solve topological problems. This thesis begins with some background material. In particular, the definition of a DCSH morphism between two coassociative chain coalgebras is recalled in complete detail. For example, signs are determined with great precision. Next we devote a chapter to the definition of Alexander-Whitney coalgebras and to their importance in topology. In the following chapter we begin the conceptual study of Alexander-Whitney coalgebras. A global study of these objects had not yet been carried out even if the Alexander-Whitney coalgebra structure has been studied and used in order to answer some specific questions. With the aim of studying Alexander-Whitney coalgebras in a nice setting, we develop an operadic description of these coalgebras in the following chapter. More precisely, we show that there is an explicit operad AW such that the coalgebras over this operad are exactly the Alexander-Whitney coalgebras. Furthermore, AW is shown to be a Hopf operad, so that the category formed by the Alexander-Whitney coalgebras is actually a monoidal category. These results are proven in a reasonably general framework. In fact, we associate an operad to each bimodule (over the associative operad) of a certain type, such that we get AW if this bimodule is well chosen. In particular, these results enable us to study Alexander-Whitney coalgebras from the standpoint of operads. This strategy is recognised to be successful in various mathematical situations, and especially in algebraic topology. Moreover, we develop a minimal model notion in the setting of right module over a chosen operad (which has to satisfy some reasonable conditions), with the aim of applying this result to the special case of the Alexander-Whitney coalgebras. This is possible because coalgebras over some fixed operad P can be seen as right modules over P. And the category of right modules over P has some nice features which do not appear to hold in the category of P-coalgebras. The inspiration for this part of our work comes from the notion of minimal model developed in the framework of rational homotopy theory. The two following facts show that it is reasonable to try to adapt some ideas of rational homotopy theory to the category of Alexander-Whitney coalgebras. A. There is a theorem that says that studying topological spaces up to rational equivalences is, essentially, equivalent to studying cocommutative chain coalgebras over the field of rational numbers. This is false if the ring of integers replaces the field of rational numbers, but Alexander-Whitney coalgebras are "almost" cocommutative in the sense which is explained in this thesis. B. It could be that the Alexander-Whitney coalgebra structure of the normalized chains of a simplicial set is weak enough to allow explicit computations. At least, it is clear that the Alexander-Whitney coalgebra structure on the normalized chains is far from being an E∞-structure (such a structure determines the homotopy type of the considered simplicial set, at least under some conditions). The chapter about minimal models in the framework of right modules over an operad includes an existence theorem and a discussion of the unicity of this model. In the second part of this chapter, we construct an explicit path-object in the model category of right modules over an operad. This path-object is then used to investigate the topologically relevant information that could stem from the minimal model in the case of the operad AW. Finally, we present and examine some interesting open questions about Alexander-Whitney coalgebras. These questions give a nice outlook on future research in this area.

EPFL2009