The homotopy category of chain complexes is a homotopy category

Homotopy Coherent Category Theory

The Homotopy Category of Complexes of Projective Modules

The homotopy category of complexes of projective left-modules over any reasonably nice ring is proved to be a compactly generated triangulated category, and a duality is given between its subcategory of compact objects and the finite derived category of right-modules.

Hom complexes and homotopy in the category of graphs

We investigate a notion of ×-homotopy of graph maps that is based on the internal hom associated to the categorical product in the category of graphs. It is shown that graph ×homotopy is characterized by the topological properties of the Hom complex, a functorial way to assign a poset (and hence topological space) to a pair of graphs; Hom complexes were introduced by Lovász and further studied ...

Hom complexes and homotopy theory in the category of graphs

We investigate a notion of ×-homotopy of graph maps that is based on the internal hom associated to the categorical product in the category of graphs. It is shown that graph ×homotopy is characterized by the topological properties of the Hom complex, a functorial way to assign a poset (and hence topological space) to a pair of graphs; Hom complexes were introduced by Lovász and further studied ...

The Homotopy Type of the Cobordism Category