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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeAug 22nd 2022
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexCreated stub <a href="https://ncatlab.org/nlab/revision/Freyd+multicategory/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Freyd+multicategory">current</a>

   Created stub

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorGuest
   • CommentTimeAug 22nd 2022
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   Author: Guest
   Format: MarkdownItexdoes the given definition violate the [[principle of equivalence]]?

   does the given definition violate the principle of equivalence?

3. • CommentRowNumber3.
   • CommentAuthormaxsnew
   • CommentTimeAug 22nd 2022
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   Author: maxsnew
   Format: MarkdownItexYes, but see the article [[identity on objects functor]] for a discussion.

   Yes, but see the article identity on objects functor for a discussion.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeAug 29th 2022
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAdd link to [[identity-on-objects functor]]. <a href="https://ncatlab.org/nlab/revision/diff/Freyd+multicategory/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Freyd+multicategory/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Freyd+multicategory">current</a>

   Add link to identity-on-objects functor.

   diff, v2, current

5. • CommentRowNumber5.
   • CommentAuthorRodMcGuire
   • CommentTimeAug 29th 2022
   • (edited Aug 29th 2022)
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItex> * A functor $return : \mathcal{V}\to\mathcal{C}$ that is the [[identity-on-objects functor|identity on objects]] and preserves central morphisms. "central morphisism" really needs to be defined here. I checked the Stanton & Levy paper, [http://www.cs.ox.ac.uk/people/samuel.staton/papers/popl13.pdf](http://www.cs.ox.ac.uk/people/samuel.staton/papers/popl13.pdf) where they define it. > If $t$ commutes with all $u$, then we say that $t$ is central. In other words, $t$ is central if it doesn’t matter when it is executed. A premulticategory is a multicategory if all morphisms are central. (Lambek [21] calls this the axiom of commutativity.) This makes the nLab definition somewhat confusing. I guess it is just saying the $return$ functor maps all morphisms of $\mathcal{V}$ to central morphisms of $\mathcal{C}$. ( the referenced Lambeck paper can be found as the first reference on our [[Joachim Lambek]] page. ) ( this distinction between [[multicategory]] and [[premulticategory]] seems to be the defining difference and really should be mentioned somewhere )

   • A functor $return : \mathcal{V}\to\mathcal{C}$ that is the identity on objects and preserves central morphisms.

   “central morphisism” really needs to be defined here. I checked the Stanton & Levy paper, http://www.cs.ox.ac.uk/people/samuel.staton/papers/popl13.pdf where they define it.

   If $t$ commutes with all $u$, then we say that $t$ is central. In other words, $t$ is central if it doesn’t matter when it is executed. A premulticategory is a multicategory if all morphisms are central. (Lambek [21] calls this the axiom of commutativity.)

   This makes the nLab definition somewhat confusing. I guess it is just saying the $return$ functor maps all morphisms of $\mathcal{V}$ to central morphisms of $\mathcal{C}$.

   ( the referenced Lambeck paper can be found as the first reference on our Joachim Lambek page. )

   ( this distinction between multicategory and premulticategory seems to be the defining difference and really should be mentioned somewhere )