The universality of the Rezk nerve

Mazel-Gee, Aaron

doi:10.2140/agt.2019.19.3217

Cited by 5 publications

(4 citation statements)

References 8 publications

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“…Let us start by recalling the model for given by the relative Rezk nerve [MG19]. For each , let us write for the subcategory of all functors and natural transformations which are pointwise egressive.…”

Section: Cartesian Fibrations Between Orthogonal Adequate Triplesmentioning

confidence: 99%

Two-variable fibrations, factorisation systems and -categories of spans

Haugseng,

Hebestreit,

Linskens

et al. 2023

Forum of Mathematics, Sigma

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We prove a universal property for $\infty $ -categories of spans in the generality of Barwick’s adequate triples, explicitly describe the cocartesian fibration corresponding to the span functor, and show that the latter restricts to a self-equivalence on the class of orthogonal adequate triples, which we introduce for this purpose. As applications of the machinery we develop, we give a quick proof of Barwick’s unfurling theorem, show that an orthogonal factorisation system arises from a cartesian fibration if and only if it forms an adequate triple (generalising work of Lanari), extend the description of dual (co)cartesian fibrations by Barwick, Glasman and Nardin to two-variable fibrations, explicitly describe parametrised adjoints (extending work of Torii), identify the orthofibration classifying the mapping category functor of an $(\infty ,2)$ -category (building on work of Abellán García and Stern), formally identify the unstraightenings of the identity functor on the $\infty $ -category of $\infty $ -categories with the (op)lax under-categories of a point, and deduce a certain naturality property of the Yoneda embedding (answering a question of Clausen).

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Section: Cartesian Fibrations Between Orthogonal Adequate Triplesmentioning

confidence: 99%

Two-variable fibrations, factorisation systems and -categories of spans

Haugseng,

Hebestreit,

Linskens

et al. 2023

Forum of Mathematics, Sigma

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“…Mazel-Gee [MG19] uses a nerve construction of Rezk to extend the localization construction to 'relative ∞-categories' in which C and W are themselves allowed to be ∞-categories. In other words, the Rezk nerve describes a very general 'calculus of fractions' for ∞-categories.…”

Section: Note That Cat Diffmentioning

confidence: 99%

“…The full definition of T as a monoidal functor between monoidal ∞-categories is rather involved and relies on a model for Cat diff ∞ based on 'relative' ∞categories; see [MG19]. The specifics of this construction are in Section 8.…”

Section: Introductionmentioning

confidence: 99%

Tangent infinity-categories and Goodwillie calculus

Bauer¹,

Burke²,

Ching³

2021

Preprint

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We make precise the analogy between Goodwillie's calculus of functors in homotopy theory and the differential calculus of smooth manifolds by introducing a higher-categorical framework of which both theories are examples. That framework is an extension to ∞-categories of the tangent categories of Cockett and Cruttwell (introduced originally by Rosický). A tangent structure on an ∞-category X consists of an endofunctor on X, which plays the role of the tangent bundle construction, together with various natural transformations that mimic structure possessed by the ordinary tangent bundles of smooth manifolds and that satisfy similar conditions.The tangent bundle functor in Goodwillie calculus is Lurie's tangent bundle for ∞-categories, introduced to generalize the cotangent complexes of André, Quillen and Illusie. We show that Lurie's construction admits the additional structure maps and satisfies the conditions needed to form a tangent ∞-category, which we refer to as the Goodwillie tangent structure on the ∞-category of ∞-categories.Cockett and Cruttwell (and others) have started to develop various aspects of differential geometry in the abstract context of tangent categories, and we begin to apply those ideas to Goodwillie calculus. For example, we show that the role of Euclidean spaces in the calculus of manifolds is played in Goodwillie calculus by the stable ∞-categories. We also show that Goodwillie's n-excisive functors are the direct analogues of n-jets of smooth maps between manifolds; to state that connection precisely, we develop a notion of tangent (∞, 2)-category and show that Goodwillie calculus is best understood in that context.

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“…where the weak equivalences in D [k] are levelwise weak equivalences, compare [Rez01, section 3.3] and [MG15,Theorem 3.8]. See also the MathOverflow post [Cis12].…”

Section: The Proof Of Theoremmentioning

confidence: 99%

Localization of cofibration categories and groupoid C∗–algebras

Land

Nikolaus

Szumiło

2017

Algebr. Geom. Topol.

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Abstract. We prove that relative functors out of a cofibration category are essentially the same as relative functors which are only defined on the subcategory of cofibrations. As an application we give a new construction of the functor that assigns to a groupoid its groupoid C * -algebra and thereby its topological K-theory spectrum.Let (C, wC, cC) be a cofibration category, i.e. a structure dual to a category of fibrant objects in the sense of Brown [Bro73]. Here, wC and cC are the subcategories of weak equivalences and cofibrations, i.e. they have the same objects as C but morphisms are the weak equivalences or the cofibrations respectively. Similarly, wcC will denote the subcategory of acyclic cofibrations. In addition to Brown's axioms, we will assume that C has good cylinders which is a mild technical condition explained in Definition 9. In this paper we will prove the following theorem. Theorem 1. If a cofibration category C has good cylinders, then the map induced by the inclusionis an equivalence of ∞-categories. In particular, by passing to homotopy categories, we obtain an equivalence of ordinary categories cC[wcBy NC[w . By passing to opposite categories, the dual statement of Theorem 1 for fibration categories also holds.The proof of Theorem 1 will be given at the end of the paper, but let us first establish a consequence and the application to C * -algebras associated to groupoids. Let C be a small cofibration category with good cylinders and M a model category which is Quillen equivalent to a combinatorial model category and has functorial fibrant and cofibrant replacements, e.g. any of the model categories of spectra. (1) F sends weak equivalences in C to weak equivalences in M.(2) F extends F in the sense that there exists a zig-zag of natural weak equivalences between F and F | cC .Moreover F is unique in the following sense: for any other functor F ′ : C → M that satisfies (1) and (2) there exists a zig-zag of natural weak equivalences between F and F ′ .Proof. We denote the ∞-category NM[w −1 ] associated to the model category M by M ∞ . We claim that for any ordinary category A the canonical mapis an equivalence of ∞-categories, where ℓ is the class of levelwise weak equivalences. If M is a simplicial, combinatorial model category this is a special case of [Lur09, Proposition 4.2.4.4] using that for a simplicial model category M, the ∞-category M ∞ is equivalent to the homotopy coherent nerve of the simplicial subcategory of M on the fibrant and cofibrant objects, see

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The universality of the Rezk nerve

Cited by 5 publications

References 8 publications

Two-variable fibrations, factorisation systems and -categories of spans

Two-variable fibrations, factorisation systems and -categories of spans

Tangent infinity-categories and Goodwillie calculus

Localization of cofibration categories and groupoid C∗–algebras

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