model category

category with weak equivalences

weak factorization system

homotopy

small object argument

resolution

Quillen adjunction

Quillen equivalence

Quillen bifunctor

derived functor

homotopy Kan extension

homotopy limit/homotopy colimit

Bousfield-Kan map

monoidal model category

enriched model category

simplicial model category

cofibrantly generated model category

combinatorial model category

cellular model category

algebraic model category

compactly generated model category

proper model category

cartesian closed model category, locally cartesian closed model category

stable model category

on functor categories (global)

on overcategories

Bousfield localization

transferred model structure

Grothendieck construction for model categories

(∞,1)-category

simplicial localization

(∞,1)-categorical hom-space

presentable (∞,1)-category

for ∞-groupoids

on topological spaces

Thomason model structure

model structure on presheaves over a test category

on simplicial sets, on semi-simplicial sets

model structure on simplicial groupoids

on cubical sets

on strict ∞-groupoids, on groupoids

on chain complexes/model structure on cosimplicial abelian groups

related by the Dold-Kan correspondence

model structure on cosimplicial simplicial sets

for n-groupoids/for n-types

for 1-groupoids

model structure on simplicial groups

model structure on reduced simplicial sets

on monoids

on simplicial T-algebras, on homotopy T-algebras

on algebas over a monad

on algebras over an operad,

on modules over an algebra over an operad

model structure on differential-graded commutative algebras

model structure on differential graded-commutative superalgebras

on dg-algebras over an operad

on dg-algebras and on on simplicial rings/on cosimplicial rings

related by the monoidal Dold-Kan correspondence

for L-∞ algebras: on dg-Lie algebras, on dg-coalgebras, on simplicial Lie algebras

model structure on dg-modules

model structure on spectra

model structure on ring spectra

model structure on presheaves of spectra

on categories with weak equivalences

Joyal model for quasi-categories

on sSet-categories

for complete Segal spaces

for Cartesian fibrations

on operads, for Segal operads

on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations

for (n,r)-categories as ∞-spaces

for weak ∞-categories as weak complicial sets

on cellular sets

on higher categories in general

on strict ∞-categories

on homotopical presheaves

on simplicial presheaves

global model structure/Cech model structure/local model structure

on simplicial sheaves

on presheaves of simplicial groupoids

on sSet-enriched presheaves

model structure for (2,1)-sheaves/for stacks

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