Two Models for the Homotopy Theory of Cocomplete Homotopy Theories

Szumiło, Karol

doi:10.48550/arxiv.1411.0303

Cited by 11 publications

(24 citation statements)

References 32 publications

(46 reference statements)

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“…Cofibration categories are models of finitely complete homotopy theories [Szu14] and admit a theory of Reedy cofibrant diagrams, with direct categories playing the role of Reedy categories to compensate for the lack of fibrations. We replicate a few definitions and lemmas that are used in the main part of the thesis; for a full discussion see [Szu14] and [RV14].…”

Section: Cofibration Categories and The Quasicategory Of Framesmentioning

confidence: 99%

“…When C is pretriangulated, the cycle category C 0 can also be seen as a cofibration category [Bro73,Sch12] which captures the homotopical structure in form of two distinguished classes of maps called weak equivalences and cofibrations. This cofibration category C 0 can in turn be made into a quasicategory N f (C 0 ) by a construction called the quasicategory of frames [Szu14], based on resolutions with homotopical Reedy cofibrant diagrams.…”

Section: Introductionmentioning

confidence: 99%

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Frames in Pretriangulated Dg-Categories

Heidemann

2020

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Triangulated categories arising in algebra can often be described as the homotopy category of a pretriangulated dg-category, a category enriched in chain complexes with a natural notion of shifts and cones that is accessible with all the machinery of homological algebra. Dg-categories are algebraic models of ∞-categories and thus fit into a wide ecosystem of higher-categorical models and translations between them. In this paper we describe an equivalence between two methods to turn a pretriangulated dg-category into a quasicategory.The dg-nerve of a dg-category is a quasicategory whose simplices are coherent families of maps in the mapping complexes. In contrast, the cycle category of a pretriangulated category forgets all higher-degree elements of the mapping complexes but becomes a cofibration category that encodes the homotopical structure indirectly. This cofibration category then has an associated quasicategory of frames in which simplices are Reedycofibrant resolutions.For every simplex in the dg-nerve of a pretriangulated dg-category we construct such a Reedy-cofibrant resolution and then prove that this construction defines an equivalence of quasicategories which is natural up to simplicial homotopy. Our construction is explicit enough for calculations and provides an intuitive explanation of the resolutions in the quasicategory of frames as a generalisation of the mapping cylinder.

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Section: Cofibration Categories and The Quasicategory Of Framesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Frames in Pretriangulated Dg-Categories

Heidemann

2020

Preprint

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“…In this section, we recall the definition of inverse categories, and define Reedy types over diagrams on inverse categories valued in a CwA. These are analogous to Reedy fibrations, a well developed tool from the homotopy theory of (co-)fibration categories [RB09,Szu14] and similar settings.…”

Section: Inverse Diagrams and Reedy Typesmentioning

confidence: 99%

“…Constructions along these lines are familiar from abstract homotopy theory, presented in terms of fibration categories [Bro73,RB09,Szu14] or comparable settings. An application of such constructions to type theory has previously been given by Shulman [Shu15], using type-theoretic fibration categories; see Remark 7.3(3) for comparison with the present work.…”

Section: Introductionmentioning

confidence: 99%

Homotopical inverse diagrams in categories with attributes

Kapulkin¹,

Lumsdaine²

2018

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We define and develop the infrastructure of homotopical inverse diagrams in categories with attributes (CwA's).Specifically, given a category with attributes C and an ordered homotopical inverse category I, we construct the category with attributes C I of homotopical diagrams of shape I in C and Reedy types over these; and we show how various logical structure (Π-types, identity types, and so on) lifts from the original CwA to the diagram CwA.This may be seen as providing a general class of diagram models of type theory, and forms a companion paper to [KL16], which applies the present results in constructing semi-model structures on categories of contextual categories.

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“…These problems can be avoided if the category with weak equivalences is known to possess more structure, namely, when it is a cofibration category (or a fibration category). Indeed, given a cofibration category C , one can associate to it its quasicategory of frames N f C , introduced by the second-named author [Szu14].…”

Section: Introductionmentioning

confidence: 99%