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Register a new userGeneral Comodule-Contramodule Correspondence
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This paper is a fundamental study of comodules and contramodules over a comonoid in a closed monoidal category. We study both algebraic and homotopical aspects of them. Algebraically, we enrich the comodule and contramodule categories over the original category, construct enriched functors between them and enriched adjunctions between the functors. Homotopically, for simplicial sets and topological spaces, we investigate the categories of comodules and contramodules and the relations between them.
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We study lax families of adjoints from a fibrational viewpoint, obtaining a version of the mate correspondence for (op)lax natural transformations of functors from an $infty$-category to the $(infty,2)$-category of $infty$-categories. We apply this to show that the left adjoint of a lax symmetric monoidal functor is oplax symmetric monoidal and that the internal Hom in a closed symmetric monoidal $infty$-category is lax symmetric monoidal in both variables. We also consider units and counits of such families of adjoints, and use them to derive the full (twisted) naturality of passing to the dual.
In this work we provide a definition of a coloured operad as a monoid in some monoidal category, and develop the machinery of Grobner bases for coloured operads. Among the examples for which we show the existance of a quadratic Grobner basis we consider the seminal Lie-Rinehart operad whose algebras include pairs (functions, vector fields).
We prove a generalisation of the correspondence, due to Resende and Lawson--Lenz, between etale groupoids---which are topological groupoids whose source map is a local homeomorphisms---and complete pseudogroups---which are inverse monoids equipped with a particularly nice representation on a topological space. Our generalisation improves on the existing functorial correspondence in four ways. Firstly, we enlarge the classes of maps appearing to each side. Secondly, we generalise on one side from inverse monoids to inverse categories, and on the other side, from etale groupoids to what we call partite etale groupoids. Thirdly, we generalise from etale groupoids to source-etale categories, and on the other side, from inverse monoids to restriction monoids. Fourthly, and most far-reachingly, we generalise from topological etale groupoids to etale groupoids internal to any join restriction category C with local glueings; and on the other side, from complete pseudogroups to ``complete C-pseudogroups, i.e., inverse monoids with a nice representation on an object of C. Taken together, our results yield an equivalence, for a join restriction category C with local glueings, between join restriction categories with a well-behaved functor to C, and partite source-etale internal categories in C. In fact, we obtain this by cutting down a larger adjunction between arbitrary restriction categories over C, and partite internal categories in C. Beyond proving this main result, numerous applications are given, which reconstruct and extend existing correspondences in the literature, and provide general formulations of completion processes.
We develop a localisation theory for certain categories, yielding a 3-arrow calculus: Every morphism in the localisation is represented by a diagram of length 3, and two such diagrams represent the same morphism if and only if they can be embedded in a 3-by-3 diagram in an appropriate way. The methods to construct this localisation are similar to the Ore localisation for a 2-arrow calculus; in particular, we do not have to use zigzags of arbitrary length. Applications include the localisation of an arbitrary model category with respect to its weak equivalences as well as the localisation of its full subcategories of cofibrant, fibrant and bifibrant objects, giving the homotopy category in all four cases. In contrast to the approach of Dwyer, Hirschhorn, Kan and Smith, the model category under consideration does not need to admit functorial factorisations. Moreover, our method shows that the derived category of any abelian (or idempotent splitting exact) category admits a 3-arrow calculus if we localise the category of complexes instead of its homotopy category.
We generalize the notion of identities among relations, well known for presentations of groups, to presentations of n-categories by polygraphs. To each polygraph, we associate a track n-category, generalizing the notion of crossed module for groups, in order to define the natural system of identities among relations. We relate the facts that this natural system is finitely generated and that the polygraph has finite derivation type.
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