ModCat (changes) in nLab

(n+1,r+1)(n+1,r+1)-categories of (n,r)-categories

• Pos

• Set

  • Rel

• Grpd, ∞Grpd

• Cat

  • Ho(Cat)

  • AccCat

  • PrCat

  • LexCat

  • MonCat

  • VCat

  • CatAdj

  • Prof

  • Operad

• 2Cat

• ModCat, CombModCat

• (∞,1)Cat

  • Pr(∞,1)Cat

  • (∞,1)Operad

• (∞,n)Cat

model category, model $\infty$-category

Definitions

• category with weak equivalences

  (relative category, homotopical category)

• fibration, cofibration

• weak factorization system

• resolution

• cell complex

• small object argument

• homotopy

• homotopy category\;of a model category

Morphisms

• Quillen adjunction

  • Quillen equivalence

  • Quillen bifunctor

  • derived functor

Universal constructions

• homotopy Kan extension

• homotopy limit/homotopy colimit

  homotopy weighted (co)limit

  homotopy (co)end

• Bousfield-Kan map

Refinements

• monoidal model category

  • monoidal Quillen adjunction

• enriched model category

  • enriched Quillen adjunction

• monoidal enriched model category

• simplicial model category

  • simplicial Quillen adjunction

  • simplicial monoidal 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

Producing new model structures

• on functor categories (global)

  • Reedy model structure

• on slice categories

• Bousfield localization

• transferred model structure

  • on algebraic fibrant objects

• Grothendieck construction for model categories

Presentation of (∞,1)(\infty,1)-categories

• (∞,1)-category

• simplicial localization

• (∞,1)-categorical hom-space

• presentable (∞,1)-category

Model structures

• Cisinski model structure

for ∞\infty-groupoids

for ∞-groupoids

• on topological spaces

  • classical model structure

  • on compactly generated spaces

  • on Delta-generated spaces

  • on diffeological spaces

  • Strøm model structure

• Thomason model structure

• model structure on presheaves over a test category

• on simplicial sets, on semi-simplicial sets

  • classical model structure

  • constructive model structure

  • for right/left fibrations

• 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 equivariant ∞\infty-groupoids

• fine model structure on topological G-spaces

• coarse model structure on topological G-spaces

  (Borel model structure)

for rational ∞\infty-groupoids

• model structure on dgc-algebras

for rational equivariant ∞\infty-groupoids

• model structure on equivariant chain complexes

• model structure on equivariant dgc-algebras

for nn-groupoids

• for n-groupoids/for n-types

• for 1-groupoids

for ∞\infty-groups

• model structure on simplicial groups

• model structure on reduced simplicial sets

for ∞\infty-algebras

general ∞\infty-algebras

• 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

specific ∞\infty-algebras

• 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

for stable/spectrum objects

• model structure on spectra

• model structure on ring spectra

• model structure on parameterized spectra

• model structure on presheaves of spectra

for (∞,1)(\infty,1)-categories

• on categories with weak equivalences

• Joyal model for quasi-categories (and its cubical version)

• on sSet-categories

• for complete Segal spaces

• for Cartesian fibrations

for stable (∞,1)(\infty,1)-categories

• on dg-categories

for (∞,1)(\infty,1)-operads

• on operads, for Segal operads

  on algebras over an operad, on modules over an algebra over an operad

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

for (n,r)(n,r)-categories

• for (n,r)-categories as ∞-spaces

• for weak ∞-categories as weak complicial sets

• on cellular sets

• on higher categories in general

• on strict ∞-categories

for (∞,1)(\infty,1)-sheaves / ∞\infty-stacks

• 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