model category

for ∞-groupoids

# Contents

## Idea

In a homotopical category $C$, a resolution of an object $X$ is another object $\stackrel{^}{X}$ equipped with a weak equivalence $\stackrel{^}{X}\to X$ or $X\to \stackrel{^}{X}$ such that $\stackrel{^}{X}$ has certain nice properties that $X$ lacks.

## In a model category

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

A fibrant resolution (or fibrant approximation) of $X$ is a fibrant object $\stackrel{^}{X}$ equipped with a weak equivalence into it

$X\stackrel{\simeq }{\to }\stackrel{^}{X}\to *\phantom{\rule{thinmathspace}{0ex}}.$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 $X$ is a cofibrant object $\stackrel{^}{X}$ equipped with a weak equivalence out of it

$\varnothing ↪\stackrel{^}{X}\stackrel{\simeq }{\to }X\phantom{\rule{thinmathspace}{0ex}}.$\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.)

## Examples

### In chain complexes

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

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

A resolution is an acyclic nonpositive complex ${P}_{\cdot }$ which coaugments $M$ or an acyclic nonnegative complex ${I}^{\cdot }$ which augments $M$, i.e. it is equipped with a map of complexes ${P}_{\cdot }\to M$ or a map of complexes $M\to {I}^{\cdot }$.

If each object ${P}_{n}$ is a projective object then ${P}_{\cdot }\to M$ is a projective resolution , and if each ${I}^{n}$ is an injective object then $M\to {I}^{\cdot }$ 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

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Revised on August 29, 2012 18:56:40 by Urs Schreiber (82.113.106.192)