Mac Lane coherence theorem

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

Let $\backslash mathsf\{Set\}\_0\; \backslash subset\; \backslash mathsf\{Set\}$ be a skeleton of the category of sets and D a unique countable set in it; note $D\; \backslash times\; D\; =\; D$ by uniqueness. Let $p\; :\; D\; =\; D\; \backslash times\; D\; \backslash to\; D$ be the projection onto the first factor. For any functions $f,\; g:\; D\; \backslash to\; D$, we have $f\; \backslash circ\; p\; =\; p\; \backslash circ\; (f\; \backslash times\; g)$. Now, suppose the natural isomorphisms $\backslash alpha:\; X\; \backslash times\; (Y\; \backslash times\; Z)\; \backslash simeq\; (X\; \backslash times\; Y)\; \backslash times\; Z$ are the identity; in particular, that is the case for $X\; =\; Y\; =\; Z\; =\; D$. Then for any $f,\; g,\; h:\; D\; \backslash to\; D$, since $\backslash alpha$ is the identity and is natural,

:$f\; \backslash circ\; p\; =\; p\; \backslash circ\; (f\; \backslash times\; (g\; \backslash times\; h))\; =\; p\; \backslash circ\; \backslash alpha\; \backslash circ\; (f\; \backslash times\; (g\; \backslash times\; h))\; =\; p\; \backslash circ\; ((f\; \backslash times\; g)\; \backslash times\; h)\; \backslash circ\; \backslash alpha\; =\; (f\; \backslash times\; g)\; \backslash circ\; p$.

Since $p$ is an epimorphism, this implies $f\; =\; f\; \backslash times\; g$. Similarly, using the projection onto the second factor, we get $g\; =\; f\; \backslash times\; g$ and so $f\; =\; g$, which is absurd.

{{section expand|date=February 2022}}

In monoidal category $C$, the following two conditions are called coherence conditions:

• Let a bifunctor $\backslash otimes\; \backslash colon\; \backslash mathbf\; C\backslash times\backslash mathbf\; C\backslash to\backslash mathbf\; C$ called the tensor product, a natural isomorphism $\backslash alpha\_\{A,B,C\}$, called the associator:

: $\backslash alpha\_\{A,B,C\}\; \backslash colon\; (A\backslash otimes\; B)\backslash otimes\; C\; \backslash rightarrow\; A\backslash otimes(B\backslash otimes\; C)$

• Also, let $I$ an identity object and $I$ has a left identity, a natural isomorphism $\backslash lambda\_A$ called the left unitor:

::$\backslash lambda\_A\; :\; I\; \backslash otimes\; A\; \backslash rightarrow\; A$

:as well as, let $I$ has a right identity, a natural isomorphism $\backslash rho\_A$ called the right unitor:

::$\backslash rho\_A\; :\; A\; \backslash otimes\; I\; \backslash rightarrow\; A$.

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.{{harvnb|Kelly|1964}}

File:Pentagonal diagram for monoidal categories.svg

File:Monoidal2.svg

• Coherency (homotopy theory)
• Monoidal category
• Symmetric monoidal category
• Coherence condition

{{reflist}}

{{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}}

{{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}}

• {{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}}

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Category:Category theory