In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism".

Often, more than one way of defining a mapping between mathematical objects might be considered "natural". Then the question might arise, which way to choose? Coherency implies that it doesn't matter which way is chosen, because all the alternative definitions are equivalent. The equivalence is often manifest in a commutative diagram.

The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g., pseudo-functor, pseudoalgebra.

In some situations, isomorphisms need to be chosen in a coherent way. Often, this can be achieved by choosing canonical isomorphisms. But in some cases, such as prestacks, there can be several canonical isomorphisms and there might not be an obvious choice among them.

In practice, coherent isomorphisms arise by weakening equalities; e.g., strict associativity may be replaced by associativity via coherent isomorphisms. For example, via this process, one gets the notion of a weak 2-category from that of a strict 2-category.

Replacing coherent isomorphisms by equalities is usually called strictification or rectification.

In a weak 2-category the composition of 1-morphisms does not satisfy associativity as an equation, however for each triple ${\displaystyle A\mathrel {\stackrel {f}{\longrightarrow }} B\mathrel {\stackrel {g}{\longrightarrow }} C\mathrel {\stackrel {h}{\longrightarrow }} D}$, there are 2-morphisms

in ${\displaystyle {\mathcal {B}}(A,D)}$ this is called associativity coherence isomorphisms.