I am studying some model categories and I recently saw the phrase "cofiber sequence". I feel like I don't really grasp what a "cofibre sequence". From what I gather if $\mathbf{M}$ is a model category and $a,b,c\in \mathbf{M}$ then $a\to b\to c$ is said to be a homotopy cofiber sequence if there is a homotopy pushout square
Is this correct? If I have a morphism $f\colon a\to b$, is the homotopy cofiber $\mathsf{cof}(f)$ of $f$ the homotopy pushout of $*\leftarrow a\xrightarrow{f} b$? If $\mathbf{M}$ is an abelian category as well do we get an exact sequence $0\to a\xrightarrow{f}b\to \mathsf{cof}(f)\to 0$ and some sort of long exact sequence? If $\mathbf{M} = \mathbf{Ch}_R$ is the category of chain complexes of $R$-modules what is the cofibre of a chain map?