In mathematics, specifically in

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topol ...

and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "

up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...

homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...

" or "up to

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

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

pseudo-functor In mathematics, a pseudofunctor ''F'' is a mapping between 2-categories, or from a category to a 2-category, that is just like a functor except that F(f \circ g) = F(f) \circ F(g) and F(1) = 1 do not hold as exact equalities but only up to ''coh ...

, pseudoalgebra.

Coherent isomorphism

In some situations, isomorphisms need to be chosen in a coherent way. Often, this can be achieved by choosing

canonical isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

s. But in some cases, such as

prestack In algebraic geometry, a prestack ''F'' over a category ''C'' equipped with some Grothendieck topology is a category together with a functor ''p'': ''F'' → ''C'' satisfying a certain lifting condition and such that (when the fibers are groupoid ...

s, 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 In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

may be replaced by associativity via coherent isomorphisms. For example, via this process, one gets the notion of a

weak 2-category In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative ''up to'' an iso ...

from that of a

strict 2-category In category theory, a strict 2-category is a category with " morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of cate ...

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

Coherence theorem

Mac Lane's coherence theorem In category theory, a branch of mathematics, Mac Lane coherence theorem states, in the words of Saunders Mac Lane, “every diagram commutes”. More precisely (cf. #Counter-example), it states every formal diagram commutes, where "formal diagram ...

states, roughly, that if diagrams of certain types commute, then diagrams of all types commute. A simple proof of that theorem can be obtained using the

permutoassociahedron In mathematics, the permutoassociahedron is an n-dimensional polytope whose vertices correspond to the bracketings of the permutations of n+1 terms and whose edges connect two bracketings that can be obtained from one another either by moving a p ...

, a

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

whose combinatorial structure appears implicitly in Mac Lane's proof. There are several generalizations of Mac Lane's coherence theorem. Each of them has the rough form that "every weak structure of some sort is equivalent to a stricter one".

Homotopy coherence

Notes

References

* * § 5. of * * Ch. 5 of * * *

External links