Ross Street

The goal is to show how a 1978 paper of Richard Wood on monoidal comonads and exponentiation relates to more recent publications such as Pastro et alia [30] and Bruguiéres et alia [7]. In the process, we mildly extend the ideas to procomonads in a magmal setting and suggest it also works for algebras for any club in the sense of Max Kelly [18, 19].

This work results from a study of Nicholas Kuhn's paper entitled "Generic representation theory of finite fields in nondescribing characteristic". Our goal is to abstract the categorical structure required to obtain an equivalence between functor categories rF , V s and rG , V s where G is the core groupoid of the category F and V is a category of modules over a commutative ring. Examples other than Kuhn's are covered by this general setting.

We denote the monoidal bicategory of two-sided modules (also called profunctors, bimodules and distributors) between categories by Mod; the tensor product is cartesian product of categories. For a groupoid G , we study the monoidal centre ZPspG , Mod op q of the monoidal bicategory PspG , Mod op q of pseudofunctors and pseudonatural transformations; the tensor product is pointwise. Alexei Davydov defined the full centre of a monoid in a monoidal category. We define a higher dimensional version: the full monoidal centre of a monoidale (= pseudomonoid) in a monoidal bicategory M , and it is a braided monoidale in the monoidal centre ZM of M. Each fibration π : H Ñ G between groupoids provides an example of a full monoidal centre of a monoidale in PspG , Mod op q. For a group G, we explain how the G-graded categorical structures, as considered by Turaev and Virelizier in order to construct topological invariants, fit into this monoidal bicategory context. We see that their structures are monoidales in the monoidal centre of the monoidal bicategory of k-linear categories on which G acts. Contents 1 The bicategory of categories and two-sided modules 5 2 Review of centres 8 3 Review of internal centres 12 4 Centre pieces in Mod op 15 5 The groupoid of automorphisms in a groupoid 17 6 Monoidales in convolution bicategories 20 7 The Turaev-Virelizier structures 22 8 Internal homs, biduals and braidings 23 9 A full centre from a groupoid fibration 25

The goal is to show how a 1978 paper of Richard Wood on monoidal comonads and exponentiation relates to more recent publications such as Pastro-Street (2009) and Bruguiéres-Lack-Virelizier (2011). In the process, we mildly extend the ideas to procomonads in a magmal setting and suggest it also works for algebras for any club in the sense of Max Kelly (1972).

The construction of a category of spans can be made in some categories A which do not have pullbacks in the traditional sense. The PROP for monoids is a good example of such an A. The 2012 book concerning homological algebra by Marco Grandis gives the proof of associativity of relations in a Puppe-exact category based on a 1967 paper of M.. Calenko. The proof here is a restructuring of that proof in the spirit of the rst sentence of this We observe that these relations are spans of EM-spans and that EM-spans admit fake pullbacks so that spans of EM-spans compose. Our setting is more general than Puppe-exact categories. We mention the formalism of distributive laws which, in a generalized form, would cover our setting. Contents 1 Suitable factorization systems 103 2 The bicategory of EM-spans 105 3 Relations as spans of spans 107 4 A fake pullback construction 107 5 An abstraction 108 6 A distributive law formalism and the PROP example 112 C are zigzag diagrams of monomorphisms and epimorphisms, rather than the jointly monomorphic spans as for a regular category (see [7] for example). Associativity of these

We go back to the roots of enriched category theory and study categories enriched in chain complexes; that is, we deal with differential graded categories (DGcategories for short). In particular, we recall weighted colimits and provide examples. We solve the 50 year old question of how to characterize Cauchy complete DG-categories in terms of existence of some specific finite absolute colimits. As well as the interactions between absolute weighted colimits, we also examine the total complex of a chain complex in a DG-category as a non-absolute weighted colimit. Contents 1 Introduction 940 2 Chain complexes 941 3 Constructions on chain complexes 944 4 DG-Categories 948 5 Module-weighted colimits 950 6 Cauchy G-modules 955 7 Cauchy DG-modules 956 8 Complexes in DG-categories 959