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
The notion of cofibration category was introduced by Hans-Joachim Baues (see references below) as a variant of the category of cofibrant objects, (for which, see category of fibrant objects and dualise). The axioms are substantially weaker than those of a Quillen model category, but add one axiom to those of K. S. Brown.
In the first chapter of his book Algebraic Homotopy (see below), Baues suggests two criteria for an axiom system:
The axioms should be sufficiently strong to permit the basic constructions of homotopy theory.
The axioms should be as weak (and as simple) as possible, so that the constructions of homotopy theory are available in as many contexts as possible.
Baues also introduces the notion of a I-category being a category with a natural cylinder functor satisfying a set of good properties.
One should distinguish cofibration categories from the Waldhausen’s notion of a “category with cofibrations and weak equivalences”, nowdays called Waldhausen category.
A cofibration category is a category $C$ with two classes of morphisms, $\mathit{cof}$ of cofibrations and $w.e.$ of weak equivalences. These are to satisfy:
C 1) Isomorphisms are both cofibrations and weak equivalences. If
are composable morphisms in $C$, then if two of $f$, $g$, $g \circ f$ are in $w.e.$, so is the third; if $f$, $g \in \mathit{cof}$ then $g \circ f\in \mathit{cof}$.
C2) Pushout axiom:
Given $B\rightarrow A$ in $\mathit{cof}$ and any $f : B\rightarrow Y$, the pushout
exists and $\bar{i}$ is in $\mathit{cof}$; if $f$ is in $w.e.$, so is $\bar{f}$; if $i$ is also in $w.e.$, so is $\bar{i}$.
C3) Factorisation axiom:
Given $B\stackrel{f}{\rightarrow} Y$, there is a factorisation
with $i \in \mathit{cof}$, $g\in w.e.$
C4) Axiom on fibrant models:
Using the terminology: “$f$ is a trivial cofibration” to mean “$f\in \mathit{cof} \cap w.e.$”, and “$RX$ is fibrant” to mean “Given any trivial cofibration $i: RX \stackrel{\simeq}{\rightarrow}Q$, there is a retraction $r : Q \rightarrow RX$, $r \circ i = Id_{RX}$”, the axiom states:
Given $X\in C$, there is trivial cofibration $X\rightarrow RX$ with $RX$ fibrant.
H.J. Baues, Algebraic homotopy, Cambridge studies in advanced mathematics 15, Cambridge University Press, (1989).