Polynomial monads and delooping of mapping spaces

Let I be a small indexing category, G: I op ! Cat be a functor and BG 2 Cat denote the Grothendieck construction on G. We define and study Quillen pairs between the category of diagrams of simplicial sets (resp. categories) indexed on BG and the category of I-diagrams over N(G) (resp. G). As an application we obtain a Quillen equivalence between the categories of presheaves of simplicial sets (resp. groupoids) on a stack M and presheaves of simplicial sets (resp. groupoids) over M. The motivation for this paper was the study of homotopy theory of (pre)sheaves on a stack. Since the site associated to a stack M is a Grothedieck construction this led us to an investigation of the homotopy theory of diagrams indexed on a category which is itself a Grothendieck construction (of a diagram of small categories). The body of the paper is concerned with analyzing various Quillen pairs between diagram categories. These adjunctions are of general interest and we present some examples not related to the theory of stacks. We conclude the paper with the applications to stacks. Stacks were introduced in algebraic geometry in order to parametrize families of objects when the presence of automorphisms prevented representability by a scheme or even a sheaf [A, DM, Gi]. Recently stacks have come to play an important role in algebraic topology. Complex oriented cohomology theories give rise to stacks over the moduli stack of formal groups and in certain situations, conversely, stacks over the moduli stack of formal groups give rise to spectra [G, R2, GHMR, B]. One fundamental example is the spectrum of topological modular forms [Hp] which is associated to the moduli stack of elliptic curves. Classically, stacks were defined as categories fibered in groupoids over a site C which satisfy descent [DM, Definition 4.1]. In [H] we show that a category fibered in groupoids F over C is a stack if and only if the assignment satisfies the homotopy sheaf

This chapter defines the Grothendieck construction for a lax functor into the category of small categories. It then proves that, for such a pseudofunctor, its Grothendieck construction is its lax colimit. Most of the rest of the chapter contains a detailed proof of the Grothendieck Construction Theorem, which states that the Grothendieck construction is part of a 2-equivalence. A generalization of the Grothendieck construction that applies to an indexed bicategory is also discussed.

Abstract2-Dimensional Categories provides an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories; pasting diagrams; lax functors; 2-/bilimits; the Duskin nerve; the 2-nerve; internal adjunctions; monads in bicategories; 2-monads; biequivalences; the Bicategorical Yoneda Lemma; and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.

Gluing derived equivalences together

The Grothendieck construction establishes an equivalence between fibrations, a.k.a. fibred categories and indexed categories and is one of the fundamental results of category theory. Cockett and Cruttwell introduced the notion of fibrations into the context of tangent categories and proved that the fibres of a tangent fibration inherit a tangent structure from the total tangent category. The main goal of this paper is to provide a Grothendieck construction for tangent fibrations. Our first attempt will focus on providing a correspondence between tangent fibrations and indexed tangent categories, which are collections of tangent categories and tangent morphisms indexed by the objects and morphisms of a base tangent category. We will show that this construction inverts Cockett and Cruttwell’s result, but it does not provide a full equivalence between these two concepts. In order to understand how to define a genuine Grothendieck equivalence in the context of tangent categories, inspired by Street’s formal approach to monad theory we introduce a new concept: tangent objects. We show that tangent fibrations arise as tangent objects of a suitable $2$ -category and we employ this characterisation to lift the Grothendieck construction between fibrations and indexed categories to a genuine Grothendieck equivalence between tangent fibrations and tangent indexed categories.

In category theory circles it is well-known that the Schreier theory of group extensions can be understood in terms of the Grothendieck construction on indexed categories. However, it is seldom discussed how this relates to extensions of monoids. We provide an introduction to the generalised Grothendieck construction and apply it to recover classifications of certain classes of monoid extensions (including Schreier and weakly Schreier extensions in particular).

The notion of an (∞,1)-category has become widely used in homotopy theory, category theory, and in a number of applications. There are many different approaches to this structure, all of them equivalent, and each with its corresponding homotopy theory. This book provides a relatively self-contained source of the definitions of the different models, the model structure (homotopy theory) of each, and the equivalences between the models. While most of the current literature focusses on how to extend category theory in this context, and centers in particular on the quasi-category model, this book offers a balanced treatment of the appropriate model structures for simplicial categories, Segal categories, complete Segal spaces, quasi-categories, and relative categories, all from a homotopy-theoretic perspective. Introductory chapters provide background in both homotopy and category theory and contain many references to the literature, thus making the book accessible to graduates and to researchers in related areas.

In this work we study the homotopy theory of coherent group actions from a global point of view, where we allow both the group and the space acted upon to vary. Using the model of Segal group actions and the model categorical Grothendieck construction we construct a model category encompassing all Segal group actions simultaneously. We then prove a global rectification result in this setting. We proceed to develop a general truncation theory for the model-categorical Grothendieck construction and apply it to the case of Segal group actions. We give a simple characterization of n-truncated Segal group actions and show that every Segal group action admits a convergent Postnikov tower built out of its n-truncations.

Category theory provides structure for the mathematical world and is seen everywhere in modern mathematics. With this book, the author bridges the gap between pure category theory and its numerous applications in homotopy theory, providing the necessary background information to make the subject accessible to graduate students or researchers with a background in algebraic topology and algebra. The reader is first introduced to category theory, starting with basic definitions and concepts before progressing to more advanced themes. Concrete examples and exercises illustrate the topics, ranging from colimits to constructions such as the Day convolution product. Part II covers important applications of category theory, giving a thorough introduction to simplicial objects including an account of quasi-categories and Segal sets. Diagram categories play a central role throughout the book, giving rise to models of iterated loop spaces, and feature prominently in functor homology and homology of small categories.

Homotopy type theory is an interpretation of constructive Martin-Lof type theory into abstract homotopy theory. It allows type theory to be used as a formal calculus for reasoning about homotopy theory, as well as more general mathematics such as can be formulated in category theory or set theory, under this new homotopical interpretation. Because constructive type theory has been implemented in computational proof assistants like Coq, it also facilitates the use of those tools in homotopy theory, category theory, set theory, and other fields of mathematics. This is the idea behind the new Univalent Foundations Program, which has recently been the object of quite intense investigation [4].

We develop the foundations of G G -global homotopy theory as a synthesis of classical equivariant homotopy theory on the one hand and global homotopy theory in the sense of Schwede on the other hand. Using this framework, we then introduce the G G -global algebraic K K -theory of small symmetric monoidal categories with G G -action, unifying G G -equivariant algebraic K K -theory, as considered for example by Shimakawa, and Schwede’s global algebraic K K -theory. As an application of the theory, we prove that the G G -global algebraic K K -theory functor exhibits the category of small symmetric monoidal categories with G G -action as a model of connective G G -global stable homotopy theory, generalizing and strengthening a classical non-equivariant result due to Thomason. This in particular allows us to deduce the corresponding statements for global and equivariant algebraic K K -theory.

Serre homotopy theory in subcategories of simplicial groups

Modelling Multi-agent Systems with Category Theory

Simple homotopy theory and nerve theorem for categories

The most recent studies in mathematics are concerned with objects, morphisms, and the relationship between morphisms. Prominent examples can be listed as functions, vector spaces with linear transformations, and groups with homomorphisms. Category theory proposes and constitutes new structures by examining objects, morphisms, and compositions. Source and target of a morphism in category theory corresponds to input and output in programming language. Thus, a connection can be obtained between category theory and functional programming languages. From this point, this paper constructs a small category implementation in a functional programming language called Haskell.