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,
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
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
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 from that of a
strict 2-category.
Replacing coherent isomorphisms by equalities is usually called strictification or rectification.
Coherence theorem
Mac Lane's coherence theorem 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, 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
See also
*
Coherence condition
*
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 ...
Notes
References
*
* § 5. of
*
* Ch. 5 of
*
*
*
External links
*https://ncatlab.org/nlab/show/homotopy+coherent+diagram
{{topology-stub
Homotopy theory