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”. This result was once thought to be the essence of the coherence theorem, but regarding a result about certain commutative diagrams, Kelly argued that, "no longer be seen as constituting the essence of a coherence theorem". More precisely (cf. #Counter-example), it states every "formal diagram", 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.

Proof

Coherence condition (Monoidal category)

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

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

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

$\lambda _{A}:I\otimes A\rightarrow A$

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

$\rho _{A}:A\otimes I\rightarrow A$.

Since there are many ways to construct an isomorphism using the above natural isomorphisms, they impose a condition called the coherence condition on the above natural isomorphisms. If the above isomorphisms satisfies the following conditions, they are called coherence conditions: the arrows constructed by get tensor products and compositions from identity morphisms, natural isomorphisms, and their inverses are equal, if their domains and codomains of the arrows are same. If the coherence condition is satisfied, then no matter how they construct the isomorphism, they will always end up with the same isomorphism. That is, the coherence conditions imply commutativity of all "formal diagrams" that one can build from the constraints.

Pentagon and triangle identity (Coherence axiom)

To satisfy the coherence condition, it is enough to prove just the pentagon identity for associativity and triangle identity for identities, which is essentially the same as what is stated in Kelly's (1964) paper.

Mac Lane coherence theorem for monoidal category

Mac Lane coherence theorem: In a monoidal category ($C,\otimes ,\alpha ,I,\lambda ,\rho$), every diagram whose vertices come from words in $\otimes$ and $I$ and whose edges come from the natural isomorphisms $\alpha ,\lambda ,\rho$ commute. To prove this theorem, it is enough to show that the pentagon and triangle identities hold.

Counter-example

It is not reasonable to expect we can show literally every diagram commutes, due to the following example of Isbell.

Let ${\mathsf {Set}}_{0}\subset {\mathsf {Set}}$ 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.

See also

• Braided monoidal category
• Coherency (homotopy theory)
• Monoidal category
• Symmetric monoidal category
• 2-category#Coherence theorem
• 3-category

References

• Eilenberg, Samuel; Kelly, G. Max (1966). "Closed Categories". Proceedings of the Conference on Categorical Algebra. pp. 421–562. doi:10.1007/978-3-642-99902-4_22. ISBN 978-3-642-99904-8.
• Kelly, G.M (1964). "On MacLane's conditions for coherence of natural associativities, commutativities, etc". Journal of Algebra. 1 (4): 397–402. doi:10.1016/0021-8693(64)90018-3.
• 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. (1986). "Braided monoidal categories" (PDF). Macquarie Math Reports. 86–0081 – via nLab.
• 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.
• MacLane, Saunders (1970). "Coherence and canonical maps". Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69). Academic Press, London-New York. pp. 231–242.
• 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.
• Johnson, Niles; Yau, Donald (2020). 2-Dimensional Categories. arXiv:2002.06055. ISBN 9780198871378.
• Johnson, Niles; Yau, Donald (2024). Bimonoidal categories, En-monoidal categories, and algebraic K-theory. Providence, Rhode Island: American Mathematical Society. ISBN 978-1-4704-7941-1.
• Schauenburg, Peter (2001). "Turning monoidal categories into strict ones". The New York Journal of Mathematics [Electronic Only]. 7: 257–265. ISSN 1076-9803.
• Loday, Jean-Louis; Vallette, Bruno (8 August 2012). Algebraic Operads. Grundlehren der mathematischen Wissenschaften. Vol. 346. Springer. doi:10.1007/978-3-642-30362-3. hdl:21.11116/0000-0004-1D0F-D. ISBN 978-3-642-30362-3.
• Power, A.J. (1989). "A general coherence result". Journal of Pure and Applied Algebra. 57 (2): 165–173. doi:10.1016/0022-4049(89)90113-8.
• Selinger, P. (2010). "A Survey of Graphical Languages for Monoidal Categories". New Structures for Physics. Lecture Notes in Physics. Vol. 813. pp. 289–355. arXiv:0908.3347. doi:10.1007/978-3-642-12821-9_4. ISBN 978-3-642-12820-2.
• Laplaza, Miguel L. (1972a). "Coherence for distributivity". Coherence in Categories. Lecture Notes in Mathematics. Vol. 281. pp. 29–65. doi:10.1007/BFb0059555. ISBN 978-3-540-05963-9.
• Laplaza, Miguel L. (1972b). "Coherence for categories with associativity, commutativity and distributivity". Bulletin of the American Mathematical Society. 78 (2): 220–222. doi:10.1090/S0002-9904-1972-12925-2.

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.
• Kelly, G. M. (1972). "An abstract approach to coherence". Coherence in Categories. Lecture Notes in Mathematics. Vol. 281. pp. 106–147. doi:10.1007/BFb0059557. ISBN 978-3-540-05963-9.

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.
• "pentagon identity". ncatlab.org.