nLab
homotopical category

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Homotopy theory

Background

Variations

Definitions

Paths and cylinders

Homotopy groups

Theorems

(∞,1)-Category theory

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

Contents

Idea

A homotopical category is a construction used in homotopy theory, related to but more flexible than a model category.

Definition

A homotopical category is a category with weak equivalences where on top of the 2-out-of-3-property the morphisms satisfy the 2-out-of-6-property:

  • If morphisms h∘g and g∘f are weak equivalences, then so are f, g, h and h∘g∘f.

Remarks

Simplicial localization

Every homotopical category C “presents” or “models” an (infinity,1)-category LC, a simplicially enriched category called the simplicial localization of C, which is in some sense the universal solution to inverting the weak equivalence up to higher categorical morphisms.

Related concepts

References

This definition is in paragraph 33 of

Revised on January 3, 2012 05:26:48 by Mike Shulman (173.8.161.189)
Edit | Back in time (10 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: Timeline of category theory and related mathematics, vertical categorification, higher category theory, Lie groupoid, omega-category, oriental, homotopy theory, anafunctor, Morita equivalence, globe, globular set, groupoid cardinality, Simpson's conjecture, (-1)-category, infinity-groupoid, semi-strict infinity-category, Cat, double category, bicategory, model category, geometric definition of higher categories, simplicial set, simplicial category, category with weak equivalences, homotopy category, derived functor, geometric shape for higher structures, cubical set, omega-groupoid, crossed complex, Nonabelian Algebraic Topology, (infinity,1)-category, 0-category, strict 2-category, periodic table, (n,r)-category, infinity-category, n-category, Kan complex, n-groupoid, k-tuply monoidal n-category, Gray-category, simplicial object, quasi-isomorphism, strict omega-category, Crans-Gray tensor product, homotopy coherent category theory, enriched homotopical category, homotopical category, category of fibrant objects, closed monoidal homotopical category, closed monoidal deformation retract, deformation retract, homotopical functor, strict omega-groupoid, model structure on topological spaces, simplex, enriched homotopy theory, simplicially enriched category, path space object, horn, simplicial complex, algebraic definition of higher categories, n-fold category, crossed module, regular epimorphism, interval object, fundamental category, tricategory, (-1)-groupoid, Kan fibration, BrownAHT, generalized universal bundle, k-surjective functor, simplicial T-complex, strict n-category, A-infinity-category, A-infinity-algebra, 2-group, small object argument, homotopy hypothesis, fibration, pseudofunctor, (infinity,n)-category, regular monomorphism, (infinity,n)-category of cobordisms, Trimble n-category, simplicial group, simplicial identities, simplicial object in Cat, simplicial groupoid, fundamental infinity-groupoid, directed homotopy theory, string diagram, homotopy 3-type, homotopy n-type, cat-n-group, 2-vector space, 2-crossed module, crossed square, double groupoid, simplicial presheaf, Dold-Kan correspondence, homotopy limit, loop space object, Grpd, 2009 January changes, delooping hypothesis, SimpSet, model structure on simplicial presheaves, Pursuing Stacks, k-tuply monoidal (n,r)-category, van Kampen theorem, higher van Kampen theorem, homotopy pullback, weak equivalence, homotopy 2-type, 3-category, homotopy type, Euler characteristic, connection on a cubical set, n-fibration, (-2)-groupoid, 1-groupoid, 1-category, 2-groupoid, 2-poset, n-poset, 3-groupoid, join of simplicial sets, ASSet, T-complex, cubical T-complex, homotopy group, fundamental group, m-cofibrant space, Postnikov system, homotopy equivalence, weak homotopy equivalence, Waldhausen category, fundamental group of a topos, Categories and Sheaves, shape theory, inverse, Higher Topos Theory, contractible space, Thomason model structure, homotopy category of an (infinity,1)-category, derived category, mapping cone, derived functor on a derived category, infinity-stackification, cohomology, Reedy category, Godement product, geometric infinity-function theory, reduced suspension, generalized Reedy category, Eckmann-Hilton argument, differential forms on simplices, calculus of fractions, associahedron, localization of an (infinity,1)-category, category of simplices, complicial set, weak complicial set, (infinity,1)-category of (infinity,1)-categories, strict 2-functor, microcosm principle, Eckmann-Hilton duality, higher category theory and physics, negative thinking, (2,1)-category, (n,1)-category, Freudenthal suspension theorem, replete subcategory, tetracategory, omega-nerve, simplicial homotopy group, boundary of a simplex, simplicial homotopy, computad, Eilenberg-Mac Lane space, fiber sequence, Galois theory, simplicial localization, n-group, transfor, k-morphism, n-functor, proper model category, generalized (Eilenberg-Steenrod) cohomology, (infinity,2)-category, n-fold complete Segal space, opetope, weak inverse, simplicial skeleton, homotopy, homotopy (as an operation), model structure on presheaves of simplicial groupoids, smooth space, exchange law, H-space, suspension, essential image, singular simplicial complex, K-theory, Gray-groupoid, lax functor, Waldhausen S-construction, simplicial resolution, Kan fibrant replacement, (infinity,1)-categorical hom-space, anodyne morphism, Batanin omega-category, differential crossed module, stable homotopy theory, derivator, homotopy - contents, higher category theory - contents, cochain on a simplicial set, free groupoid, n-truncated object of an (infinity,1)-category, A-infinity-space, A1-homotopy theory, thin homotopy, cellular set, homotopy localization, spine, infinitesimal interval object, Whitehead tower, dendroidal set, Whitehead theorem, hypercomplete (infinity,1)-topos, quasifibration, suspension object, cocylinder, Hurewicz cofibration, mapping cylinder, Theta-space, Theta category, deformation retract of a homotopical category, homotopy extension property, homotopy inverse, nPOV, mapping cocone, internal infinity-groupoid, Whitehead product, companion pair, cocycle, simplicial ring, cofibration, coherence theorem for monoidal categories, covering, Grothendieck-Maltsiniotis infinity-category, automorphism infinity-group, primary homotopy operation, (infinity,n)-module, categorical homotopy groups in an (infinity,1)-topos, simplicial topological group, applications of (higher) category theory, 2Cat, CW approximation, equivariant homotopy theory, Homotopy Theory of Higher Categories, test category, free loop space object, model structure on cubical sets, semicategory, simplicial localization of a homotopical category, (infinity,1)-category theory, Calculus of fractions and homotopy theory, homotopy coherent diagram, decalage, hypergroupoid, clique, coherence theorem, Ho(Top), hypercrossed complex, proper homotopy theory, homotopy Kan extension, vertical composition, globular operad, pasting diagram, associator, unitor, pointed derivator, zigzag, algebraic Kan complex, algebraic quasi-category, subdivision, 2-category theory, two-out-of-six property, automorphism 2-group, homotopy exact square, pullback in a derivator, nice simplicial topological space, geometric homotopy groups in an (infinity,1)-topos, infinite loop space, embedding, effective monomorphism, Duskin nerve, cosmic cube, homotopy equivalence of toposes, symmetric monoidal (infinity,n)-category, n-connected space, long exact sequence of homotopy groups, universal principal infinity-bundle, resolution, Gray group, factorization lemma, hom-groupoid, horizontal composition, augmented simplicial set, holographic principle of higher category theory, cosimplicial simplicial set, fundamental (infinity,1)-category, infinity-anafunctor, 3-morphism, coherence law, associativity, 1-morphism, (1,1)-category, pentagonator, hexagonator, strict 2-groupoid, bigroupoid, zigzag category, essential fiber, geometric realization of cohesive infinity-groupoids, chain homotopy, model structure on simplicial algebras, homotopy T-algebra, nGrpd, (infinity,n)-presheaf, higher homotopy van Kampen theorem, cell complex, blob n-category, model structure on operator algebras, fundamental infinity-groupoid of a locally infinity-connected (infinity,1)-topos, van Kampen theorem for toposes, fundamental infinity-groupoid in a locally infinity-connected (infinity,1)-topos, product of simplices, Pi-algebra, spherical object, model structure on algebras over an operad, (infinity,n)Cat, Boardman-Vogt resolution, Brown-Grossman homotopy group, integral transforms on sheaves, integral transform, (infinity,n)-category of spans, localizing subcategory, classical triangulation, geometric realization of categories, fully dualizable object, Betti number, homotopy groups of a Lie groupoid, cohomology operation, (infinity,2)-sheaf, natural equivalence, two-out-of-three, (infinity,n)-category with duals, Homotopy Limit Functors on Model Categories and Homotopical Categories, semi-simplicial set, homotopy type theory, Parametrized Homotopy Theory, looping, truncated object, dendroidal homotopy coherent nerve, bisimplicial set, contents of contents, B1-homotopy theory, descent object, homotopy groups of spheres, suspension of a chain complex, Hurewicz theorem, outer automorphism infinity-group, homotopy level, (infinity,2)-topos, 0-morphism, function extensionality, coend in a derivator, univalence axiom, algebraic model category, Bousfield lattice, model structure on cellular sets, parity complex, Cartan-Eilenberg category, elegant Reedy category, higher Segal space, Simplicial homotopy theory, simplicial homotopy theory, iterated loop space, Cartesian fibration of dendroidal sets, (n,r)-congruence, Segal n-category, Eilenberg subcomplex, model structure for infinity-groupoids, homotopy 2-category, local fibration, reduced simplicial set, projective resolution, (infinity,n)-sheaf, coherence theorem for symmetric monoidal categories, null homotopy, interval object in chain complexes, (infinity,n)-category with adjoints, homotopy class, homotopy category of chain complexes, Ho(sSet), free resolution, n-connected map, homotopical structure on C*-algebras, TwoVect, h-groupoid, model topos, coherence theorem for braided monoidal categories, coherence theorem for monoidal bicategories, coherence theorem for braided monoidal bicategories, coherence theorem for symmetric monoidal bicategories, (infinity,n)-topos, symmetric monoidal 2-category, localization of model categories, globular theory, model structure for homotopy n-types, Toda bracket, differential Galois theory, semi-simplicial object, core in a 2-category, double Lie groupoid, Lie algebroid-groupoid, relative category