Anodinmuodot

Anodinmuodot (anodyne morphisms) are a concept in abstract homotopy theory and model category theory. In a model category, a morphism f: A → B is called anodinmuoto if it has the left lifting property with respect to every fibration. Equivalently, anodinmuodot are the maps that can be obtained from a generating set of morphisms by pushouts, compositions, and retracts; they form the left class in the standard two-class factorization together with fibrations and weak equivalences.

In many model structures, anodinmuodot are closely related to cofibrations and weak equivalences: they are often the acyclic cofibrations (cofibrations that are also weak equivalences). A typical generating set for anodyne morphisms in the model category of simplicial sets is given by the horn inclusions Λ^n_k → Δ^n for 0 ≤ k ≤ n and n ≥ 1.

Properties commonly cited for anodinmuodot include closure under pushouts, transfinite compositions, and retracts. They behave as “harmless” maps in the sense that they do not alter the homotopy type as detected by fibrations, which makes them useful in constructing factorizations and in proving lifting results.

Origins and terminology: the concept arose with the development of Quillen’s model categories in the 1960s and 1970s. The term “anodyne” conveys the idea that these morphisms are harmless to homotopy theory. In Finnish mathematical literature the term is rendered as anodinmuodot, aligning with the general terminology for morphism classes in model categories.