Given an equivalence of categories $C \equiv D$, such that $C$ has a monoidal structure, is it clear that we can use the equivalence to induce a monoidal structure on $D$. Is there a standard reference for this?
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3$\begingroup$ Yes, it is clear. I doubt the details are written explicitly anywhere, but it is a special case of a much more general result about pseudo-algebras for a 2-monad (also, sadly, not in the literature). $\endgroup$– Zhen LinCommented May 27, 2014 at 19:49
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1$\begingroup$ If I recall correctly the result about transport of pseudoalgebra structure can be found in ``Monoidal functors generated by adjunctions, with applications to transport of structure." by Kelly and Lack. I can't get my hands of the paper but the review on MathSciNet puts it as Prop 6.1 of that paper. I doubt this is great as a standard reference for that fact because it is quite a basic thing, but studied there as a special case of something more complex. It might have appeared earlier too. $\endgroup$– johnCommented May 28, 2014 at 12:02
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$\begingroup$ I would say that this follows from a very general principle that every good structure in category theory transports via equivalences of categories. Here, good means that we are not allowed to say "nonsense" such as that two objects are equal (but only isomorphic via some specific isomorphism). $\endgroup$– Martin BrandenburgCommented Jun 3, 2014 at 12:00
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2$\begingroup$ This is a basic exercise, so it needs no reference. $\endgroup$– S. Carnahan ♦Commented Jun 13, 2014 at 0:34
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Like Zhen Lin, I too doubt the existence of literature on this topic. However, let me provide the usual construction.
Equivalences of categories are defined such that they preserve every structure and property defined in category theory. Suppose $F^\prime: \mathcal{D}\to \mathcal{C}$ is a quasi-inverse to $F$. Define a monoidal structure on $\mathcal{D}$ as $X \otimes Y := F(G(X) \otimes G(Y))$ and $1_\mathcal{D} := F(1_\mathcal{C})$. You could check the associativity and the unit constraint, induced by $\mathcal{C}$. It can be observed that $F$ becomes a monoidal equivalence between $\mathcal{C}$ and $\mathcal{D}$.
Added: As Qiaochu says in the comments, "it shouldn't be true that you can transport a strict monoidal structure to another strict monoidal structure along an equivalence of categories". As Martin Brandenburg says here, strict monoidal categories are not definable in the (proper) language of categories. (Strict monoidal categories belong instead in set theory.)
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5$\begingroup$ The first statement should be false if not interpreted carefully. For example, it shouldn't be true that you can transport a strict monoidal structure to another strict monoidal structure along an equivalence of categories, although I don't know an explicit counterexample. The analogous statement in homotopy theory is that you shouldn't be able to transport a strictly associative multiplication to another strictly associative multiplication along a homotopy equivalence. So there is really something to check here, namely that the usual definition of a monoidal category is weak enough. $\endgroup$ Commented Jun 12, 2014 at 21:03
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$\begingroup$ @QiaochuYuan Strict monoidal categories are not definable in the (proper) language of categories. (Strict monoidal categories belong instead in set theory.) See my edit incorporating that into the answer. $\endgroup$– user62675Commented Jun 12, 2014 at 22:40
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4$\begingroup$ (1) You're quoting Martin Brandenburg. (2) I have yet to see an explicit definition of the so-callled "proper language of categories". $\endgroup$– Zhen LinCommented Jun 13, 2014 at 1:02
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$\begingroup$ @ZhenLin Yes, I found Martin Brandenburg's comment here. I forgot to cite it; I apologize for that. I have added it into the answer. Again, I apologize for forgetting to cite it. $\endgroup$– user62675Commented Jun 13, 2014 at 1:07