View source for Mac Lane coherence theorem

In [[category theory]], a branch of mathematics, '''Mac Lane's coherence theorem''' states, in the words of [[Saunders Mac Lane]], “every diagram commutes”.<ref>{{harvnb|Mac Lane|1998|loc=Ch VII, § 2.}}</ref> But regarding a result about certain commutative diagrams, Kelly is states as follows: "no longer be
seen as constituting the essence of a coherence theorem".<ref>{{harvnb|Kelly|1974|loc=1.2}}</ref> More precisely (cf. [[#Counter-example]]), it states every [[formal diagram]] commutes, where "formal diagram" is an analog of well-formed formulae and terms in [[proof theory]].

The theorem can be stated as a strictification result; namely, every [[monoidal category]] is monoidally equivalent to a strict monoidal category.<ref>{{harvnb|Schauenburg|2001}}</ref>

== Counter-example ==
It is ''not'' reasonable to expect we can show literally every diagram commutes, due to the following example of Isbell.<ref>{{harvnb|Mac Lane|1998|loc=Ch VII. the end of § 1.}}</ref>

Let <math>\mathsf{Set}_0 \subset \mathsf{Set}</math> be a [[Skeleton (category theory)|skeleton]] of the category of sets and ''D'' a unique [[countable set]] in it; note <math>D \times D = D</math> by uniqueness. Let <math>p : D = D \times D \to D</math> be the projection onto the first factor. For any functions <math>f, g: D \to D</math>, we have <math>f \circ p = p \circ (f \times g)</math>. Now, suppose the natural isomorphisms <math>\alpha: X \times (Y \times Z) \simeq (X \times Y) \times Z</math> are the identity; in particular, that is the case for <math>X = Y = Z = D</math>. Then for any <math>f, g, h: D \to D</math>, since <math>\alpha</math> is the identity and is natural,
:<math>f \circ p = p \circ (f \times (g \times h)) = p \circ \alpha \circ (f \times (g \times h)) = p \circ ((f \times g) \times h) \circ \alpha = (f \times g) \circ p</math>.
Since <math>p</math> is an epimorphism, this implies <math>f = f \times g</math>. Similarly, using the projection onto the second factor, we get <math>g = f \times g</math> and so <math>f = g</math>, which is absurd.

== Proof ==
{{section expand|date=February 2022}}
=== Coherence condition (Monoidal category) ===
In monoidal category <math>C</math>, the following two conditions are called [[coherence condition]]s:

*Let a [[bifunctor]] <math>\otimes \colon \mathbf C\times\mathbf C\to\mathbf C</math> called the [[tensor product]], a [[natural isomorphism]] <math>\alpha_{A,B,C}</math>, called the ''associator'':

: <math>\alpha_{A,B,C} \colon (A\otimes B)\otimes C \rightarrow A\otimes(B\otimes C)</math>

*Also, let <math>I</math> an identity object and <math>I</math> has a left identity, a natural isomorphism <math>\lambda_A </math> called the ''left unitor'':

::<math>\lambda_A : I \otimes A \rightarrow A </math>

:as well as, let <math>I</math> has a right identity, a natural isomorphism <math>\rho_A </math> called the ''right unitor'':

::<math>\rho_A : A \otimes I \rightarrow A </math>.

===Pentagon and triangle identity===
To satisfy the coherence condition, it is enough to prove just the pentagon and triangle identity, which is essentially the same as what is stated in Kelly's (1964) paper.<ref>{{harvnb|Kelly|1964}}</ref>
[[File:Pentagonal diagram for monoidal categories.svg]]
[[File:Monoidal2.svg]]

==See also==
*[[Coherency (homotopy theory)]]
*[[Monoidal category]]
*[[Symmetric monoidal category]]
*[[Coherence condition]]

== Notes ==
{{reflist}}

== References ==
{{ref begin}}
*{{cite journal |doi=10.1016/0021-8693(64)90018-3 |title=On MacLane's conditions for coherence of natural associativities, commutativities, etc |date=1964 |last1=Kelly |first1=G.M |journal=Journal of Algebra |volume=1 |issue=4 |pages=397–402 }}
*{{cite journal |doi=10.1017/S0960129508007184 |title=On traced monoidal closed categories |date=2009 |last1=Hasegawa |first1=Masahito |journal=Mathematical Structures in Computer Science |volume=19 |issue=2 |pages=217–244 }}
*{{cite journal |doi=10.1006/aima.1993.1055 |doi-access=free |title=Braided Tensor Categories |date=1993 |last1=Joyal |first1=A. |authorlink1=André Joyal |last2=Street |first2=R. |authorlink2=Ross Street |journal=[[Advances in Mathematics]] |volume=102 |issue=1 |pages=20–78 }}
* {{cite journal |url=https://hdl.handle.net/1911/62865 |hdl=1911/62865 |title=Natural Associativity and Commutativity |date=October 1963 |last1=MacLane |first1=Saunders |journal=Rice Institute Pamphlet - Rice University Studies }}
* {{cite journal |doi=10.1090/S0002-9904-1965-11234-4 |title=Categorical algebra |date=1965 |last1=MacLane |first1=Saunders |journal=Bulletin of the American Mathematical Society |volume=71 |issue=1 |pages=40–106 |doi-access=free }}
* {{cite book | last=Mac Lane | first=Saunders | title=Categories for the working mathematician | publisher=Springer | publication-place=New York | date=1998 | isbn=0-387-98403-8 | oclc=37928530|url={{Google books|MXboNPdTv7QC|page=165|plainurl=yes}}}}
*Section 5 of Saunders Mac Lane, {{cite journal |doi=10.1090/S0002-9904-1976-13928-6 |title=Topology and logic as a source of algebra |date=1976 |last1=Mac Lane |first1=Saunders |journal=Bulletin of the American Mathematical Society |volume=82 |issue=1 |pages=1–40 |doi-access=free }}
*{{cite journal |url=http://eudml.org/doc/121925 |title=Turning monoidal categories into strict ones |journal=The New York Journal of Mathematics [Electronic Only] |date=2001 |volume=7 |pages=257–265 |last1=Schauenburg |first1=Peter|issn=1076-9803}}
{{ref end}}

==Further reading==
{{ref begin}}
*{{cite book |doi=10.1007/BFb0063106 |chapter=Coherence theorems for lax algebras and for distributive laws |title=Category Seminar |series=Lecture Notes in Mathematics |date=1974 |last1=Kelly |first1=G. M. |volume=420 |pages=281–375 |isbn=978-3-540-06966-9 }}
{{ref end}}

== External links ==
*{{cite web |last1=Armstrong |first1=John |title=Mac Lane's Coherence Theorem |url=https://unapologetic.wordpress.com/2007/06/29/mac-lanes-coherence-theorem/ |website=The Unapologetic Mathematician|date=29 June 2007 }}
*{{cite web |last1=Etingof |first1=Pavel |last2=Gelaki |first2=Shlomo |last3=Nikshych |first3=Dmitri |last4=Ostrik |first4=Victor |title=18.769, Spring 2009, Graduate Topics in Lie Theory: Tensor Categories §.Lecture 3 |url=https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/resources/mit18_769s09_lec03/ |website=MIT Open Course Ware}}
*{{cite web|title=coherence theorem for monoidal categories |url=https://ncatlab.org/nlab/show/coherence+theorem+for+monoidal+categories |website=ncatlab.org}}
*{{cite web|title=Mac Lane's proof of the coherence theorem for monoidal categories |url=https://ncatlab.org/nlab/show/Mac+Lane%27s+proof+of+the+coherence+theorem+for+monoidal+categories |website=ncatlab.org}}
*{{cite web|title=coherence and strictification|url=https://ncatlab.org/nlab/show/coherence+and+strictification|website=ncatlab.org}}
*{{cite web|title=coherence and strictification for monoidal categories|url=https://ncatlab.org/nlab/show/coherence+and+strictification+for+monoidal+categories|website=ncatlab.org}}
*{{cite web|title=pentagon identity|url=https://ncatlab.org/nlab/show/pentagon+identity|website=ncatlab.org}}
{{categorytheory-stub}}


[[Category:Category theory]]