A complete and cocomplete category with designated model weak equivalences, model fibrations and model cofibrations, closed under retracts. Weak equivalences satisfy two-out-of-three; maps admit the two cofibration/fibration factorizations, and either acyclic class has the required lifting property against the opposite class. These axioms organize homotopy-theoretic calculations abstractly.
Attach cells for all lifting squares at each successive stage, then take a suitable colimit. Smallness of the generating domains makes every eventual lifting square occur at an earlier stage. This factors a map into a relative cell complex followed by a map with the prescribed right lifting property.
A composite built by successive pushouts of coproducts of prescribed generating maps. In nonnegative chain complexes, attaching a sphere-to-disk generator adds a free element with a specified existing cycle as boundary. The resulting inclusion has a degreewise free cokernel.
Mapping out of the object preserves sequential colimits. A map factors through a finite stage, and two finite-stage maps agreeing in the colimit agree at a later stage. Relative sequential smallness restricts the diagrams to a stated class of morphisms.
The model category of nonnegative complexes of left modules has quasi-isomorphisms as model weak equivalences, positive-degree surjections as model fibrations, and degreewise injections with projective cokernel as model cofibrations. All objects are fibrant; cofibrant objects are degreewise projective.
The sphere chain complex inclusions into disk chain complexes, together with the separate degree-zero cell, generate the model cofibrations by cell attachment and retracts. Their right-injective class is exactly the acyclic fibration in the projective chain-complex model structure.
The right lifting property for each sphere-to-disk map fills a prescribed cycle and a compatible target element. The degree-zero generator supplies initial surjectivity; all other fillers force an acyclic kernel and the remaining degreewise surjections.
A map that is both a model fibration and a quasi-isomorphism. The homology isomorphism together with degree-one surjectivity forces surjectivity in degree zero. Equivalently it has the right lifting property against the generating cofibrations for nonnegative chain complexes.
A morphism in the chosen cofibration class of a model category. In the projective model structure on nonnegative chain complexes it is a degreewise split injection with degreewise projective cokernel. This usage is distinguished from the topological cofibration.
A morphism in the chosen fibration class of a model category. In the projective model structure on nonnegative chain complexes, it is surjective in strictly positive degrees; degree zero is not required.
A morphism in the chosen weak-equivalence class of a model category. In the projective model structure on nonnegative chain complexes these are quasi-isomorphisms.
Articles by others on the same topic
A **model category** is a concept from category theory, which is a branch of mathematics that deals with abstract structures and relationships between them. Specifically, a model category provides a framework for doing homotopy theory in a categorical setting. It allows mathematicians to work with "homotopical" concepts such as homotopy equivalences, fibrations, and cofibrations in a systematic way.