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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 10th 2014

added to polynomial functor the evident but previously missing remark why it is called a “polynomial”, here.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeSep 12th 2016

I added some remarks to polynomial functor: “being polynomial” is a mere property of a (strong) functor once its domain and codomain are identified with slices of an ambient category, cartesian transformations between polynomial functors can be identified on the polynomial data, and the bicategory of polynomial functors enhances to a double category.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeSep 12th 2016

I made what I thought were similar remarks at the section on polynomial endofunctors at tree, in particular about a double category structure. Could you please have a look, Mike?

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeSep 12th 2016

Yes, this is all in Gambino-Kock, right? I guess maybe they don’t explicitly write down a version of the double category that contains only cartesian cells.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeSep 12th 2016

I don’t remember seeing Gambino-Kock; I just took a guess based on what was in Kock’s article referenced in tree.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeSep 12th 2016

here’s the link; the double category is in section 3.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeApr 20th 2020

It would be nice to mention some of the results in this paper in the body too, but I don’t have time.

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeMar 15th 2021

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeMar 15th 2021

• Workshop on Polynomial Functors, Topos Institute, 15–19 March 2021, (website)
1. Fixed a broken link in a reference.

Rongmin Lu

• CommentRowNumber11.
• CommentAuthorDavidRoberts
• CommentTimeMar 7th 2023

Added the example of how a “literal polynomial” endofunctor of a lextensive category really is a polynomial functor (here), with a sketch proof.

And included a little discussion of how this almost works for more general extensive categories, except one can really only talk about the composite $\Pi_{\nabla_{Y_n}}\circ (Y\sqcup Y)^*$, as the dependent product $\Pi_{\nabla_{Y_n}}$ itself fails to exist in the absence of all pullbacks.

Also mentioned the version with a “literal power series” functor in the case that countable coproducts exist.

• CommentRowNumber12.
• CommentAuthorDavidRoberts
• CommentTimeMar 17th 2023
• (edited Mar 17th 2023)

Actually, if we are happy to permit a partial right adjoint to pullback along the copairing map $\nabla_A\colon A\sqcup A \to A$, namely only defined for those objects of $\mathcal{C}/(A\sqcup A)$ (isomorphic to those) of the form $A\times B_2 \sqcup A\times B_2 \to A\sqcup A$, then a literal polynomial makes sense in any extensive category with finite products and can be seen as a polynomial endofunctor using the partially defined dependent product $\Pi_{\nabla_A}$ functor.