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 the context of homotopy theory modeled by homotopical categories, where equality is relaxed to a weaker concept of (weak) homotopy equivalence or generally to weak equivalences, a resolution of any object $X$ is a choice of weak equivalence to or from another object $X_{res}$, such that $X_{res}$ has some prescribed good properties which $X$ itself may be lacking.
This general concept may be formalized by homotopical categories that carry extra information on what counts as a “good property” of an object, called fibrant objects or cofibrant objects. This includes fibration categories/cofibration categories and notably model categories.
A common cause for need of resolutions occurs when in a category of nice objects certain universal constructions – such as quotients or intersections – fail to exist. Homotopical resolution may allow to embed the nice objects in categories of systems of nice objects (such as simplicial objects or chain complexes) inside which homotopical resolutions may be found (such as simplicial resolutions or homological resolutions, respectively).
For example the quotient of a scheme or a smooth manifold by the action of an algebraic group or Lie group may fail to be a scheme or smooth manifold again, respectively, but the corresponding action groupoid (orbifold) serves as a resolution of the quotient in a suitable homotopical category of simplicial objects in schemes/manifolds (presenting the “quotient stack”, see at higher geometry for more).
Dually the intersection of two subschemes/submanifolds may fail to be a scheme/manifold itself, but the corresponding derived intersection? may provide a resolution in a homotopical category of cosimplicial objects in schemes/manifolds, respectively (see at derived geometry for more).
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 homological resolutions in one of the standard model structures 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.
See at simplicial resolution.