algebraic model category


Model category theory

model category



Universal constructions


Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras



for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

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

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



The structure of an algebraic model category is a refinement of that of a model category.

Where a bare model category structure is a category with weak equivalences refined by two weak factorization systems ((cofibrations,acyclicfibrations)(cofibrations, acyclic fibrations) and (acycliccofibrations,fibrations)(acyclic cofibrations, fibrations)) in an algebraic model structure these are refined further to algebraic weak factorization systems plus a bit more.

This extra structure supplies more control over constructions in the model category. For instance its choice induces a weak factorization system also in every diagram category of the given model category.


An algebraic model structure on a homotopical category (M,W)(M,W) consists of a pair of algebraic weak factorization systems (C t,F)(C_t, F), (C,F t)(C,F_t) together with a morphism of algebraic weak factorization systems

(C t,F)(C,F t)(C_t,F) \to (C,F_t)

such that the underlying weak factorization systems form a model structure on MM with weak equivalences WW.

A morphism of algebraic weak factorization systems consists of a natural transformation

domf C tf Cf Rf ξ f Qf Ff F tf codf \array{ & \text{dom} f & \\ {}^{C_{t}f}\swarrow & & \searrow {}^{{C}{f}} \\ Rf & \stackrel{\xi_f}{\to} & Qf \\ {}_{{F}{f}}\searrow & & \swarrow {}_{F_{t}f} \\ & \text{cod} f & }

comparing the two functorial factorizations of a map ff that defines a colax comonad morphism C tCC_t \to C and a lax monad morphism F tFF_t \to F.


Every cofibrantly generated model category structure can be lifted to that of an algebraic model category.

Any algebraic model category has a fibrant replacement monad RR and a cofibrant replacement comonad QQ. There is also a canonical distributive law RQQRRQ \to QR comparing the two canonical bifibrant replacement functors.


The notion was introduced in

The algebraic analog of monoidal model categories is discussed in

Revised on January 9, 2012 22:13:32 by Emily Riehl (