nLab
enriched Quillen adjunction

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Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for ∞-groupoids

for n-groupoids

for ∞-groups

for ∞-algebras

general

specific

for stable/spectrum objects

for (∞,1)-categories

for stable (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for (∞,1)-sheaves / ∞-stacks

Enriched category theory

enriched category theory

Background

Basic concepts

Universal constructions

Extra stuff, structure, property

Homotopical enrichment

Contemts

Definition

In enriched model category theory, an enriched Quillen adjunction is an adjunction in the sense of enriched category theory whose underlying ordinary adjunction is a Quillen adjunction between ordinary model categories.

Here “underlying” refers to the underlying ordinary category C 0 of any V-enriched category, defined by C 0(x,y)=V(I,C(x,y)). (Recall that an enriched model category is an enriched category, together with a model structure on its underlying ordinary category, and some compatibility conditions.)

Special cases

A special role is playes by sSet-enriched Quillen adjunctions, for the standard model structure on simplicial sets. See simplicial Quillen adjunction for more on that

Revised on August 25, 2010 15:13:45 by Urs Schreiber (131.211.36.96)
Edit | Back in time (4 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: canonical model structure, hypercover, model category, category with weak equivalences, homotopy category, derived functor, model structure on simplicial sets, category of fibrant objects, model structure on topological spaces, monoidal model category, path space object, pushout-product axiom, BrownAHT, small object argument, homotopy limit, model structure on simplicial presheaves, model structure on simplicial sheaves, homotopy pullback, Thomason model structure, Reedy category, generalized Reedy category, Bousfield-Kan map, simplicial model category, enriched model category, model structure on strict omega-groupoids, spectral sequence, proper model category, Quillen equivalence, local model structure on simplicial presheaves, global model structure on simplicial presheaves, Quillen adjunction, model structure on presheaves of simplicial groupoids, model structure on sSet-enriched presheaves, model structure on chain complexes, combinatorial model category, cofibrantly generated model category, homotopy image, Reedy model structure, enriched Reedy category, model structure on functors, Kan fibrant replacement, (infinity,1)-categorical hom-space, model structure for Cartesian fibrations, model structure on dg-algebras, Quillen bifunctor, model category theory - contents, model structure on dendroidal sets, tractable model category, model structure on homotopical presheaves, cellular model category, model structure on an over category, Bousfield localization of model categories, model structure on dg-coalgebras, canonical model structure on Cat, model structure for complete Segal spaces, model structure for quasi-categories, model structure on algebraic fibrant objects, model structure on sSet-categories, stable model category, model structure on cubical sets, model structure for left fibrations, model structure on operads, transferred model structure, homotopy Kan extension, Cisinski model structure, fat simplex, monoid axiom in a monoidal model category, resolution, factorization lemma, simplicial Quillen adjunction, model structure on cosimplicial abelian groups, model structure on monoids in a monoidal model category, trivial model structure, model structure on simplicial algebras, model structure on reduced simplicial sets, model structure on algebras over a monad, model structure on dg-Lie algebras, cell complex, model structure on simplicial groupoids, model structure on operator algebras, monoidal Quillen adjunction, model structure on algebras over an operad, model structure on spectra, model structure on modules over an algebra over an operad, Boardman-Vogt resolution, hyper-derived functor, model structure on strict omega-categories, model structure on simplicial groups, model structure on dg-operads, model structure on dg-algebras over an operad, model structure on dg-categories, canonical model structure on groupoids, model structure for (2,1)-sheaves, model structure on dg-modules, E-infinity operad, model structure for L-infinity algebras, model structure on presheaves over a test category, model structure on cosimplicial simplicial sets, compactly generated model category, contents of contents, locally presentable categories - introduction, model structure on sSet-operads, model structure for Segal operads, generalized Reedy model structure, cartesian closed model category, model structure for Segal categories, model structure on simplicial Lie algebras, excellent model category, algebraic model category, 2-trivial model structure, locally cartesian closed model category, model structure on presheaves of spectra, fibrant object, model structure on cellular sets, model structure for dendroidal left fibrations, model structure for dendroidal Cartesian fibrations, elegant Reedy category, model structure for dendroidal complete Segal spaces, canonical model structure on Operad, projectively cofibrant diagram, model structure for infinity-groupoids, fibrant replacement, model structure on enriched categories, model topos, localization of model categories, model structure for homotopy n-types, relative category, model structure on semi-simplicial sets