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”. 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". 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.

Counter-example

It is not reasonable to expect we can show literally every diagram commutes, due to the following example of Isbell. Let ${\mathsf{S}\mathsf{e}\mathsf{t}}_0\subset \mathsf{S}\mathsf{e}\mathsf{t}$ be a skeleton of the category of sets and D a unique countable set in it; note $D\times D=D$ by uniqueness. Let $p:D=D\times D\to D$ be the projection onto the first factor. For any functions $f,g:D\to D$, we have $f\circ p=p\circ (f\times g)$. Now, suppose the natural isomorphisms $\alpha :X\times (Y\times Z)\simeq (X\times Y)\times Z$ are the identity; in particular, that is the case for $X=Y=Z=D$. Then for any $f,g,h:D\to D$, since $\alpha$ is the identity and is natural,

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

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

Proof

Coherence condition

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

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

${\alpha }_{A,B,C}:(A\otimes B)\otimes C\to A\otimes (B\otimes C)$

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

${\lambda }_A:I\otimes A\to A$

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

${\rho }_A:A\otimes I\to A$.

Pentagon identity and triangle identity

File:Pentagonal diagram for monoidal categories.svg File:Monoidal2.svg

See also

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

References

• Hasegawa, Masahito (2009). "On traced monoidal closed categories". Mathematical Structures in Computer Science. 19 (2): 217–244. doi:10.1017/S0960129508007184.
• Joyal, A.; Street, R. (1993). "Braided Tensor Categories". Advances in Mathematics. 102 (1): 20–78. doi:10.1006/aima.1993.1055.
• MacLane, Saunders (October 1963). "Natural Associativity and Commutativity". Rice Institute Pamphlet - Rice University Studies. hdl:1911/62865.
• MacLane, Saunders (1965). "Categorical algebra". Bulletin of the American Mathematical Society. 71 (1): 40–106. doi:10.1090/S0002-9904-1965-11234-4.
• Mac Lane, Saunders (1998). Categories for the working mathematician. New York: Springer. ISBN 0-387-98403-8. OCLC 37928530.
• Section 5 of Saunders Mac Lane, Mac Lane, Saunders (1976). "Topology and logic as a source of algebra". Bulletin of the American Mathematical Society. 82 (1): 1–40. doi:10.1090/S0002-9904-1976-13928-6.
• Schauenburg, Peter (2001). "Turning monoidal categories into strict ones". The New York Journal of Mathematics [Electronic Only]. 7: 257–265. ISSN 1076-9803.

Further reading

• Kelly, G. M. (1974). "Coherence theorems for lax algebras and for distributive laws". Category Seminar. Lecture Notes in Mathematics. Vol. 420. pp. 281–375. doi:10.1007/BFb0063106. ISBN 978-3-540-06966-9.

External links

• Armstrong, John (29 June 2007). "Mac Lane's Coherence Theorem". The Unapologetic Mathematician.
• Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. "18.769, Spring 2009, Graduate Topics in Lie Theory: Tensor Categories §.Lecture 3". MIT Open Course Ware.
• "coherence theorem for monoidal categories". ncatlab.org.
• "Mac Lane's proof of the coherence theorem for monoidal categories". ncatlab.org.
• "coherence and strictification". ncatlab.org.
• "coherence and strictification for monoidal categories". ncatlab.org.