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I am attempting to read "Tensor Categories" by Pavel Etingof et. al.

The following pentagon axiom is a part of the definition of a monoidal category:

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The following diagram is part of the definition of a monoidal functor:

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My $1^{\text{st}}$ question is

How do I intuitively think about the meaning of diagrams like these? For example, what information are either of these diagrams trying to convey and why are they required as part of the definitions?

On a similar note, one proof in the book goes like this: "consider the following diagram:"

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Another proof goes like this: "Consider the diagram:"

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My $2^{\text{nd}}$ question is

Is there any rhyme or reason to "considering" these diagrams, or are these just tricks I need to learn and get comfortable to the point where considering some large diagram like this becomes automatic? Also, the same question as above applies here: is there any intuitive way of thinking about these diagrams that can make everything simpler?

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    $\begingroup$ Commutative diagrams are just a visual representation of equations. If you are more comfortable with equations you can rewrite them as systems of equations. $\endgroup$
    – Zhen Lin
    Commented Jun 4, 2023 at 4:53
  • $\begingroup$ By an equation, for example, for the first diagram above, you mean $a_{W,X,Y\otimes Z}\circ a_{W\otimes X, Y, Z}=1_W\otimes a_{X,Y,Z}\circ a_{W,X\otimes Y, Z}\circ a_{W,X,Y}\otimes 1_Z$? Or did you mean some equation that involves "classical" objects - like a relation between elements of a monoid that the monoidal category is categorifying? $\endgroup$ Commented Jun 4, 2023 at 15:22

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One of the reasons behind considering the pentagonal axiom and monoidal structure axiom is to assert the coherence theorem for monoidal categories. Basically, given an ordered sequence $(X_1,....,X_n)$ of objects in a monoidal category, there are many ways to parenthesise the n-fold tensor product $X_1 \otimes X_2 \otimes ... \otimes X_n)$. Of course, the existence of the associator gives that any two such parenthesized products will be isomorphic. But one could have different isomorphisms between them, so that you could have different identifications for the two products. The coherence theorem says that in fact, any two such identifications are going to be the same. For $n=3$, there are exactly two ways of parenthesizing $X_1 \otimes X_2 \otimes X_3$. There is exactly one canonical isomorphism here, namely the component of the associator at $(X_1,X_2,X_3)$. The pentagonal axiom certainly makes sure that this is the case when $n = 4$. Finally, the monoidal structure axiom closes the argument for all $n$. Precisely, it can be shown that for every monoidal category $(\mathcal(C), \otimes, 1,...)$, there is a monoidal equivalence from $\mathcal{C}$ into a strict monoidal category. The monoidal structure axiom can then be used to prove the coherence theorem in general by mapping a diagram (consisting of two isomorphisms of parenthesized products of some ordered sequence of objects) into a strict monoidal category along a monoidal equivalence, and then reflect back the commutativity of the "image" to assert the commutativity of the original diagram.

Strictly speaking, the coherence axioms always appear when there is some sort of categorification going on. Categorification is sort of a dictionary that translates notions from algebra to notions in category theory. Elements of an algebraic structure become objects in a category and equations which are part of the defining axioms for the algebraic structure become natural isomorphisms. But one expects these isomorphisms to behave in a consistent or coherent fashion. This is where the coherence axioms come into play.

A monoidal category categorifies the notion of a Monoid in algebra. A monoidal functor categorifies the concept of monoid homomorphisms. The coherence axioms in this situation are the Pentagonal axiom, Unit axiom, Monoidal structure axioms etc.

The difficult task is to figure out what would the right coherence axioms be in other instances of categorification.

Regarding you second question, yes, the ability to play around with large commutative diagrams certainly improves over time. As mentioned in the comments, you can always write down the equation that you need to prove and then translate it into the language of commutative diagrams (which categorify equations).

Have fun!

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