# Coherency (homotopy theory)

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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".

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.

## Coherent isomorphism

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.

## Coherence theorem

The Mac Lane coherence theorem states, roughly, that if diagrams of certain types commute, then diagrams of all types commute.

There are several generalizations (see for instance [1]). But each such a theorem has the rough form that “every weak structure of some sort is equivalent to a stricter one”.[1]

## Notes

1. ^ Shulman, 1. Introduction

## References

• Cordier, J.M., and T. Porter. "Homotopy coherent category theory." Trans. Amer. Math. Soc. 349 (1), 1997, 1–54.
• § 5. of Mac Lane, Saunders, Topology and Logic as a Source of Algebra (Retiring Presidential Address), Bulletin of the AMS 82:1, January 1976.
• Mac Lane, Saunders (1971). Categories for the working mathematician. Graduate texts in mathematics Springer-Verlag. Especially Chapter VII Part 2.
• Ch. 5 of K. H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory
• Shulman, Mike (2012). "Not every pseudoalgebra is equivalent to a strict one". Adv. Math. 229 (3): 2024–2041. arXiv:1005.1520.