Coherency (homotopy theory)

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.