Lallement Functor is a Weak Right Multiadjoint

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.

The Grothendieck construction for model categories

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.

Gluing derived equivalences together

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).

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

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.

Weak fibrations

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.

In this paper we provide purely categorical proofs of two important results of structural Ramsey theory: the result of M. Sokić that the free product of Ramsey classes is a Ramsey class, and the result of M. Bodirsky that adding constants to the language of a Ramsey class preserves the Ramsey property. The proofs that we present here ignore the model-theoretic background of these statements. Instead, they focus on categorical constructions by which the classes can be constructed generalizing the original statements along the way. It turns out that the restriction to classes of relational structures, although fundamental for the original proof strategies, is not relevant for the statements themselves. The categorical proofs we present here remove all restrictions on the signature of first-order structures and provide the information not only about the Ramsey property but also about the Ramsey degrees.

Support varieties for transporter category algebras

We show how to apply forward and reverse mode Combinatory Homomorphic Automatic Differentiation (CHAD) (Vákár (2021). ESOP, 607–634; Vákár and Smeding (2022). ACM Transactions on Programming Languages and Systems44 (3) 20:1–20:49.) to total functional programming languages with expressive type systems featuring the combination of •tuple types;•sum types;•inductive types;•coinductive types;•function types.We achieve this by analyzing the categorical semantics of such types in $\Sigma$ -types (Grothendieck constructions) of suitable categories. Using a novel categorical logical relations technique for such expressive type systems, we give a correctness proof of CHAD in this setting by showing that it computes the usual mathematical derivative of the function that the original program implements. The result is a principled, purely functional and provably correct method for performing forward- and reverse-mode automatic differentiation (AD) on total functional programming languages with expressive type systems.

We extend some classical results – such as Quillen’s Theorem A, the Grothendieck construction, Thomason’s theorem and the characterisation of homotopically cofinal functors – from the homotopy theory of small categories to polynomial monads and their algebras. As an application we give a categorical proof of the Dwyer–Hess and Turchin results concerning the explicit double delooping of spaces of long knots.

This paper explores the relationship amongst the various simplicial and pseudo-simplicial objects characteristically associated to any bicategory C. It proves the fact that the geometric realizations of all of these possible candidate `nerves of C' are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space BC of the bicategory. Its other major result proves a direct extension of Thomason's `Homotopy Colimit Theorem' to bicategories: When the homotopy colimit construction is carried out on a diagram of spaces obtained by applying the classifying space functor to a diagram of bicategories, the resulting space has the homotopy type of a certain bicategory, called the `Grothendieck construction on the diagram'. Our results provide coherence for all reasonable extensions to bicategories of Quillen's definition of the `classifying space' of a category as the geometric realization of the category's Grothendieck nerve, and they are applied to monoidal (tensor) categories through the elemental `delooping' construction.

Let G G be a finite group. Over any finite G G -poset P \mathcal {P} we may define a transporter category G ∝ P G\propto \mathcal {P} as the corresponding Grothendieck construction. There exists a Becker-Gottlieb transfer from the ordinary cohomology of G ∝ P G\propto \mathcal {P} to that of G G . We shall construct it using module-theoretic methods and then extend it to a transfer from the Hochschild cohomology of k ( G ∝ P ) k(G\propto \mathcal {P}) to that of k G kG , where k k is a base field.