This entry is about resolutions in the sense of homotopy theory. For resolutions of singularities see at resolution of singularities.


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



In a homotopical category CC, a resolution of an object XX is another object X^\hat X equipped with a weak equivalence X^X\hat X \to X or XX^X \to \hat X such that X^\hat X has certain nice properties that XX lacks.

See also simplicial resolution.

In a model category

If CC is a model category then the most important resolutions are cofibrant resolutions and fibrant resolutions.

A fibrant resolution (or fibrant approximation) of XX is a fibrant object X^\hat X equipped with a weak equivalence into it

XX^*. X \stackrel{\simeq}{\to} \hat X \to * \,.

If the weak equivalence is also a cofibration, the fibrant resolution is a good fibrant resolution.

A cofibrant resolution (or cofibrant approximation) of XX is a cofibrant object X^\hat X equipped with a weak equivalence out of it

X^X. \emptyset \hookrightarrow \hat X \stackrel{\simeq}{\to} X \,.

If the weak equivalence is also a fibration the cofibrant resolution is a good cofibrant resolution.

Notice that the factorization axioms of a model category ensure that such resolutions always exist.

Of course for the notion of fibrant resolution to make sense, also the ambient structure of a category of fibrant objects works. For cofibrant resolutions a Waldhausen category does the job, etc.

In the context of cofibration categories, the term used is fibrant model. (One also finds the term fibrant replacement used.)


In chain complexes

We consider the case of the one of the standard model structure on chain complexes.

If CC is a category of chain complexes in a suitable (possibly structured) abelian category or semiabelian category AA then one can in particular consider resolutions of ordinary objects of AA – regarded as a chain complex concentrated in degree 0 - by chain complexes of AA.

A resolution is an acyclic nonpositive complex P P_\cdot which coaugments MM or an acyclic nonnegative complex I I^\cdot which augments MM, i.e. it is equipped with a map of complexes P MP_\cdot \to M or a map of complexes MI M\to I^\cdot.

If each object P nP_n is a projective object then P MP_\cdot \to M is a projective resolution , and if each I nI^n is an injective object then MI M\to I^\cdot is an injective resolution . These are fibrant and cofibrant resolutions in the suitable model structure on chain complexes.

There are further generalizations like unbounded resolutions etc.

In (co)simplicial objects in a model category


Revised on August 11, 2015 06:25:55 by Urs Schreiber (