Talk:Monoidal functor

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The map from I to FI in the definition is just a morphism (which may, of course, be viewed as a natural transformation between constant functors, but that seems a bit overkill).

I think I am going to have to change it myself if no one else will. What do you think?

It doesn't seem overkill to me, in fact, that's the only way I could make sense of it. Otherwise, it's a “type error” in my head, akin to using a scalar where a 1x1 matrix is expected.

What doesn't make sense to me is that one of the coherence maps is a (not necessarily invertible) morphism, but the other is a natural isomorphism. Shouldn't the latter be a (not necessarily invertible) natural transformation? Eduardo León (talk) 22:57, 16 September 2016 (UTC)Reply

Elsewhere on the Internet, references seem only to use "monoidal functor" to refer to what this article calls "strong monoidal functors" [and, I must admit, my most natural interpretation of "monoidal functor" before looking it up would have had it preserve tensor products (up to isomorphism, etc.)]. E.g., the description here: [1]. Are these two competing popular usages? -Chinju (talk) 06:25, 8 July 2008 (UTC)Reply

Yes, Chinju, they are two competing usages. What's described here is the terminology of the Australian school of category theory, following Kelly. While that terminology has been taken up quite widely, there are many people who say "lax monoidal functor" for what is called "monoidal functor" here, and "monoidal functor" for what is called "strong monoidal functor" here. I'll make an edit to reflect that. 86.0.206.49 (talk) 16:10, 21 December 2008 (UTC)Reply

I think the page could still use more clarity in this respect. As written, it defines "monoidal functors" and makes vague references to "lax monoidal functors", but never explicitly mentions they are the same. The differences in terminology that user 86.0.206.49 mentions should probably be be spelt out explicitly in the article. Nathaniel Virgo (talk) 05:22, 14 August 2020 (UTC)Reply

For now I've edited the definition section so that it defines "lax monoidal functor" instead of "monoidal functor", which then agrees with what's written in the lede. (I've no preference about which term is used, but if "monoidal functor" is preferred it would mean editing the lede to explain that.) Nathaniel Virgo (talk) 05:28, 14 August 2020 (UTC)Reply