Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeDec 26th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea A refinement of the notion of [[homotopy category]] of a [[model category]] (more generally, [[relative category]]) that keeps track of homotopies between morphisms. Dwyer–Kan implement the notion of a simplicial homotopy category as a simplicial category that is Dwyer–Kan equivalent to the [[hammock localization]]. ## Related concepts * [[homotopy category]] * [[category of fractions]] * [[hammock localization]] ## References See §5.1 in * {#DwyerKan80FunctionComplexes} [[William G. Dwyer]], [[Daniel M. Kan]], _Function complexes in homotopical algebra_, Topology 19 (1980), 427-440 &lbrack;<a href="https://doi.org/10.1016/0040-9383(80)90025-7">doi:10.1016/0040-9383(80)90025-7</a>, [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/dwyer-kan-3.pdf)&rbrack; <a href="https://ncatlab.org/nlab/revision/simplicial+homotopy+category/1">v1</a>, <a href="https://ncatlab.org/nlab/show/simplicial+homotopy+category">current</a>

   Created:

   Idea

   A refinement of the notion of homotopy category of a model category (more generally, relative category) that keeps track of homotopies between morphisms.

   Dwyer–Kan implement the notion of a simplicial homotopy category as a simplicial category that is Dwyer–Kan equivalent to the hammock localization.

   Related concepts

   • homotopy category

   • category of fractions

   • hammock localization

   References

   See §5.1 in

   • {#DwyerKan80FunctionComplexes} William G. Dwyer, Daniel M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427-440 [doi:10.1016/0040-9383(80)90025-7, pdf]

   v1, current