Coherency (homotopy theory)

In mathematics, specifically in homotopy theory, it is typically necessary to have equalities or diagrams that hold “up to homotopy”. But, for theory to make sense, “up to homotopy” cannot be done arbitrarily but has to be in some coherent way.

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

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 there can be several canonical isomorphisms and there might not be an obvious choice among them; prestack for example for this type of the issue.

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 MacLane coherence theorem states, roughly, that if diagrams of certain types commute, then diagrams of all types commute.

There are several generalizations; see for instance https://ncatlab.org/nlab/show/coherence+theorem

See also

• Coherence condition
• Canonical isomorphism

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

External links

• https://ncatlab.org/nlab/show/homotopy+coherent+diagram