nLab homotopy coherent category theory (changes)

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**category theory** ## Concepts {#sidebar_concepts} * category * functor * natural transformation * Cat ## Universal constructions {#sidebar_universal_constructions} * universal construction * representable functor * adjoint functor * limit/colimit * weighted limit * end/coend * Kan extension ## Theorems {#sidebar_theorems} * Yoneda lemma * Isbell duality * Grothendieck construction * adjoint functor theorem * monadicity theorem * adjoint lifting theorem * Tannaka duality * Gabriel-Ulmer duality * small object argument * Freyd-Mitchell embedding theorem * relation between type theory and category theory ## Extensions {#sidebar_extensions} * sheaf and topos theory * enriched category theory * higher category theory ## Applications {#sidebar_applications} * applications of (higher) category theory

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*** **homotopy theory, (∞,1)-category theory, homotopy type theory** flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed... models: topological, simplicial, localic, ... see also **algebraic topology** **Introductions** * Introduction to Basic Homotopy Theory * Introduction to Abstract Homotopy Theory * geometry of physics -- homotopy types **Definitions** * homotopy, higher homotopy * homotopy type * Pi-algebra, spherical object and Pi(A)-algebra * homotopy coherent category theory * homotopical category * model category * category of fibrant objects, cofibration category * Waldhausen category * homotopy category * Ho(Top) * (∞,1)-category * homotopy category of an (∞,1)-category **Paths and cylinders** * left homotopy * cylinder object * mapping cone * right homotopy * path object * mapping cocone * universal bundle * interval object * homotopy localization * infinitesimal interval object **Homotopy groups** * homotopy group * fundamental group * fundamental group of a topos * Brown-Grossman homotopy group * categorical homotopy groups in an (∞,1)-topos * geometric homotopy groups in an (∞,1)-topos * fundamental ∞-groupoid * fundamental groupoid * path groupoid * fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos * fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos * fundamental (∞,1)-category * fundamental category **Basic facts** * fundamental group of the circle is the integers **Theorems** * fundamental theorem of covering spaces * Freudenthal suspension theorem * Blakers-Massey theorem * higher homotopy van Kampen theorem * nerve theorem * Whitehead's theorem * Hurewicz theorem * Galois theory * homotopy hypothesis-theorem

Homotopy coherent category theory or enriched homotopy theory is the attempt to understand those situations that arise in homotopy theory, homotopical algebra, and non-Abelian homological algebra, in which quite naturally occuring diagrams are not commutative, yet are commutative ‘up-to-homotopy’.

If the diagram is just commutative in the homotopy category, that is not much use and one can do little with it. Surprisingly often however, the homotopies involved in such a diagram’s ‘almost commutative’ nature can be specified, but then the question arises as to whether those homotopies form some sort of diagram, so homotopies between composite homotopies become involoved. This begins to look like parts of 2-category theory, or rather ‘higher weak category theory’ and the development of homotopy coherent category theory was initiated in an attempt to merge homotopy theory with categorical tools for handling higher categories.

In particular: *~~homotopy theory~~ ~~using~~ ~~model categories~~ and similar structures; * ~~enriched category theory~~.

In the case that the category $V$ one is enriching over is itself a model category or at least a category with weak equivalences one wishes to generalize limits and in particular the weighted limits such as as ends and coends in enriched category theory to constructions which satisfy the familiar universal properties only up to coherent homotopy.

homotopy theory using model categories and similar structures;
enriched category theory.

~~References~~

References

These articles deal with the theory of homotopy coherent diagrams:

R. Vogt, Homotopy limits and colimits , Math. Z., 134, (1973), 11 – 52.
J.-M. Cordier and T. Porter, Vogt’s theorems on categories of homotopy coherent diagrams, Math. Proc. Camb. Phil. Soc. 100 (1986), 65–90.
J.-M. Cordier and T. Porter, Maps between homotopy coherent diagrams, Top. and its Applications, 28 (1988) 255-275.
J.-M. Cordier and T. Porter, Fibrant diagrams, rectifications and a construction of Loday, J. Pure Appl. Alg 67 (1990), 111–124.

A discussion of homotopy limits is in

D. Bourn and J.-M. Cordier, A general formulation of homotopy limits , J. Pure Appl. Algebra, 29, (1983), 129–141,
J.-M. Cordier, Sur les limites homotopiques de diagrammes homotopiquement cohérents, Comp. Math. 62 (1987), 367–388.

J-M Cordier and T. Porter, Homotopy coherent category theory, Trans. Amer. Math. Soc. 349 (1997) 1-54. (web)

a main point is the definition and discussion of a homotopy coherent end? for the case of enrichment over the model category of simplicial sets.

Mike Shulman, Homotopy limits and colimits and enriched homotopy theory (arXiv)

the general issue of enriched homotopy theory is addressed and enriched homotopical categories are introduced, which are a coherent combination of the notion of enriched category with that of homotopical category

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