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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 3rd 2010
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   Author: Urs
   Format: MarkdownItexthe entry [[monoidal functor]] did not state the axioms. I put them in.

   the entry monoidal functor did not state the axioms. I put them in.

2. • CommentRowNumber2.
   • CommentAuthorTobias Fritz
   • CommentTimeJan 24th 2013
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   Author: Tobias Fritz
   Format: MarkdownItexHmm, it seems that the entry defines what a monoidal functor is only between strict monoidal categories. I'm thinking about editing the page so that it covers the general case. I suppose it might be helpful to state the definition in the strict case first before going to the general and more complex situation. Any thoughts? By the way, has anybody written down a coherence theorem for monoidal functors? As in: given two monoidal cats $C$ and $D$ and a monoidal functor $F:C\to D$, every (formal) diagram in $D$ constructed from the given data commutes? (I'm pretty sure that this is true, but haven't actually proved it.)

   Hmm, it seems that the entry defines what a monoidal functor is only between strict monoidal categories. I’m thinking about editing the page so that it covers the general case. I suppose it might be helpful to state the definition in the strict case first before going to the general and more complex situation. Any thoughts?

   By the way, has anybody written down a coherence theorem for monoidal functors? As in: given two monoidal cats $C$ and $D$ and a monoidal functor $F:C\to D$, every (formal) diagram in $D$ constructed from the given data commutes? (I’m pretty sure that this is true, but haven’t actually proved it.)

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 24th 2013
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   Author: Todd_Trimble
   Format: MarkdownItexTobias, see also [[pseudofunctor]]. Thinking of a monoidal category as a 1-object bicategory, one could just refer to that page for the definition of monoidal functor, if one didn't feel like writing out the diagrams. As for coherence of monoidal functors: see Coherence for Tricategories by Gordon-Power-Street, page 4 for a description and for references. (Let me know if you have trouble accessing that. I think it's a good idea to enter this into the Lab, and I could volunteer myself to do that unless you want to do this.)

   Tobias, see also pseudofunctor. Thinking of a monoidal category as a 1-object bicategory, one could just refer to that page for the definition of monoidal functor, if one didn’t feel like writing out the diagrams.

   As for coherence of monoidal functors: see Coherence for Tricategories by Gordon-Power-Street, page 4 for a description and for references. (Let me know if you have trouble accessing that. I think it’s a good idea to enter this into the Lab, and I could volunteer myself to do that unless you want to do this.)

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJan 25th 2013
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   Author: Mike Shulman
   Format: MarkdownItexI wonder whether there is some abstract approach to this. E.g. now that we have Bourke's 2-monadicity theorem, we can relatively easily identify lax monoidal functors with lax $T$-morphisms for a 2-monad $T$; is there a general coherence theorem for lax morphisms between algebras for a 2-monad?

   I wonder whether there is some abstract approach to this. E.g. now that we have Bourke’s 2-monadicity theorem, we can relatively easily identify lax monoidal functors with lax $T$-morphisms for a 2-monad $T$; is there a general coherence theorem for lax morphisms between algebras for a 2-monad?

5. • CommentRowNumber5.
   • CommentAuthorTobias Fritz
   • CommentTimeJan 26th 2013
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   Author: Tobias Fritz
   Format: HtmlI was just about to edit <a href="http://ncatlab.org/nlab/show/monoidal%20functor">monoidal functor</a> accordingly, but now I see that Urs already did that on Thursday :)

   I was just about to edit monoidal functor accordingly, but now I see that Urs already did that on Thursday :)