Kan extensions and the calculus of modules for ∞–categories

Riehl, Emily; Verity, Dominic

doi:10.2140/agt.2017.17.189

Cited by 15 publications

(11 citation statements)

References 10 publications

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“…Every model category has an underlying derivator [5,6] and therefore a good calculus of homotopy (co)limits. The same is true for a bicomplete (∞, 1)-category, as shown by Riehl and Verity in [20]. Now, for a relative category, we might be able to construct homotopy (co)limits but not a derivator structure.…”

Section: Introductionmentioning

confidence: 78%

Double homotopy (co)limits for relative categories

Werndli¹

2018

An Alpine Bouquet of Algebraic Topology

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We answer the question to what extent homotopy (co)limits in categories with weak equivalences allow for a Fubini-type interchange law. The main obstacle is that we do not assume our categories with weak equivalences to come equipped with a calculus for homotopy (co)limits, such as a derivator.Proposition 8.1. Let C be a category and J an index category. For J ∈ J with inclusion In J : {J} ֒→ J, the evaluation functor ev J : C J → C has a right adjoint(with the obvious arrow map), granted all these powers in C exist. Moreover, if End J (J) = {id J }, then J * is fully faithful or equivalently, a counit ε of the adjunction is invertible. Dually, ev J has a left adjointgranted all these copowers exist and if End J (J) = {id J }, then J ! is fully faithful.Proof. Note that ev J = J * is given by precomposition with In J : {J} ֒→ J and thus has a right adjoint given by taking the pointwise right Kan extension along J (granted that all these exist)Example 8.2. For J any direct or inverse category (in the sense of the theory of Reedy categories), then J ! and J * (assuming they exist) are fully faithful for all J ∈ J. In particular, this is true for the indexing categories of the most common homotopy colimits, such as homotopy pushouts, coproducts, telescopes and coequalisers.

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Section: Introductionmentioning

confidence: 78%

Double homotopy (co)limits for relative categories

Werndli¹

2018

An Alpine Bouquet of Algebraic Topology

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“…Following Wood [Woo82] and Grandis and Paré [GP08], who used 'proarrow equipments' and double categories respectively to formalise parts of classical category theory, recently certain unital virtual double categories have been used to study formal category theory in less well behaved settings, as follows. Cruttwell and Shulman in [CS10] internalise the notion of fully faithful morphism in the unital virtual double category Mod(X) of 'modules' in a virtual double category X, while Riehl and Verity in [RV17] internalise the notions of fully faithful morphism, 'exact square' and (pointwise) Kan extension in the unital virtual double category Mod K of modules between ∞-categories in the homotopy 2-category of a '∞-cosmos' K.…”

Section: Motivation: Internalising Yoneda Embeddingsmentioning

confidence: 99%

Augmented virtual double categories

Koudenburg¹

2019

Preprint

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In this article the notion of virtual double category [Cruttwell-Shulman, 2010] (also known as (fc-)multicategory [Burroni, 1971;Leinster, 2004]) is extended as follows. While cells in a virtual double category classically have a horizontal multi-source and single horizontal target, the notion of augmented virtual double category introduced here extends the latter notion by including cells with empty horizontal target as well.Any augmented virtual double category comes with a built-in notion of "locally small object" and we describe advantages of using augmented virtual double categories as a setting for formal category rather than 2-categories, which are classically equipped with a notion of "admissible object" by means of a yoneda structure [Street-Walters, 1978].An object is locally small precisely if it admits a horizontal unit, and we show that the notions of augmented virtual double category and virtual double category coincide in the presence of all horizontal units. Without assuming the existence of horizontal units we show that most of the basic theory for virtual double categories, such as that of restriction and composition of horizontal morphisms, extends to augmented virtual double categories. We introduce and study in augmented virtual double categories the notion of "pointwise" composition of horizontal morphisms, which formalises the classical composition of profunctors given by the coend formula.

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“…Specialising to the case of quasi-categories, the structures thus defined are equivalent to those introduced in the work of Lurie [12] and Joyal [9] respectively. A companion paper applies this 2-categorical theory in certain slice categories to obtain a notion of two-sided groupoidal cartesian fibrations upon which the calculus of modules (profunctors) between ∞-categories will be founded [20].…”

Section: Cartesian Fibrationsmentioning

confidence: 99%

Fibrations and Yoneda's lemma in an ∞-cosmos

Riehl

Verity

2017

Journal of Pure and Applied Algebra

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We use the terms ∞-categories and ∞-functors to mean the objects and morphisms in an ∞-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of fibrant objects. Quasi-categories, Segal categories, complete Segal spaces, marked simplicial sets, iterated complete Segal spaces, θ n -spaces, and fibered versions of each of these are all ∞-categories in this sense. Previous work in this series shows that the basic category theory of ∞-categories and ∞-functors can be developed only in reference to the axioms of an ∞-cosmos; indeed, most of the work is internal to the homotopy 2-category, a strict 2-category of ∞-categories, ∞-functors, and natural transformations. In the ∞-cosmos of quasi-categories, we recapture precisely the same category theory developed by Joyal and Lurie, although our definitions are 2categorical in natural, making no use of the combinatorial details that differentiate each model.In this paper, we introduce cartesian fibrations, a certain class of ∞-functors, and their groupoidal variants. Cartesian fibrations form a cornerstone in the abstract treatment of "category-like" structures a la Street and play an important role in Lurie's work on quasi-categories. After setting up their basic theory, we state and prove the Yoneda lemma, which has the form of an equivalence between the quasi-category of maps out of a representable fibration and the quasi-category underlying the fiber over its representing element. A companion paper will apply these results to establish a calculus of modules between ∞-categories, which will be used to define and study pointwise Kan extensions along ∞-functors.

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Kan extensions and the calculus of modules for ∞–categories

Cited by 15 publications

References 10 publications

Double homotopy (co)limits for relative categories

Double homotopy (co)limits for relative categories

Augmented virtual double categories

Fibrations and Yoneda's lemma in an ∞-cosmos

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