This entry is about resolutions in the sense of homotopy theory. For resolutions of singularities see at resolution of singularities.
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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
In a homotopical category $C$, a resolution of an object $X$ is another object $\hat X$ equipped with a weak equivalence $\hat X \to X$ or $X \to \hat X$ such that $\hat X$ has certain nice properties that $X$ lacks.
See also simplicial resolution.
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 $\hat X$ equipped with a weak equivalence into it
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 $\hat X$ equipped with a weak equivalence out of it
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.)
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.
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