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Stream: event: Polynomial Functors @ Topos

Topic: David Spivak: "The polynomial abacus" 1

Tim Hosgood (Mar 15 2021 at 14:26):

Monday, 21:00 UTC

Tim Hosgood (Mar 15 2021 at 20:59):

this is the next talk, and will be starting very shortly

Joachim Kock (Mar 15 2021 at 21:06):

First in a series of two talks:

Abstract: A set-theoretic polynomial functor is naturally a combinatorial object, which might be imagined as an abacus: a collection of columns, each equipped with a set of beads. Moving quickly but with little outside knowledge beyond the Yoneda lemma, I will use these abacuses to very concretely tell the story of P, the framed bicategory of polynomial comonads and bicomodules. In particular, I'll discuss both theory and applications of P.

For theory I'll begin with an overview of some pleasing properties enjoyed by the category of polynomials. Then I'll explain Ahman-Uustalu's result that—up to isomorphism—polynomial comonads are categories and Garner's result that the bicomodules between them are parametric right adjoints between the associated copresheaf categories. In particular every copresheaf topos is found as a hom-category in P. I'll also discuss the cofree comonad construction.

For applications, I'll recall the notion of Moore machines and show that they are objects in a copresheaf topos C-Set for a particular category C; I'll explain how the story generalizes to arbitrary C. I'll also explain how databases and data migration functors live naturally in this setting. Time permitting I may also discuss how deep learning and cellular automata show up in P.

Tim Hosgood (Mar 15 2021 at 21:19):

in the notation of Poly, the Yoneda map satisfies $y^y=y+1$, which implies that "the Yoneda embedding is equal to $\approx 1.77678$"... I wonder if this is a nice magic number somehow

Steve Awodey (Mar 15 2021 at 21:51):

this equation is known from Lawvere's "objective number theory". It holds for example for the generic object in the object classifying topos Set^FIN, and related categories.

Eigil Rischel (Mar 15 2021 at 21:59):

At the beginning of the talk, @David Spivak mentioned that $\mathbf{Poly}$ can equivalently be expressed as

• The full subcategory of $[Set,Set]$ spanned by coproducs of (co)representables
• The full subcategory of $[Set,Set]$ spanned by those functors that preserve connected limits.

Any coproduct of corepresentables, beyond preserving connected limits, will also be accessible, i.e preserve $\kappa$-filtered colimits for $\kappa$ sufficiently large (I think taking $\kappa$ the cardinality of the union of all the corepresenting sets will suffice, but I am not 100% on this). So I think the correct statement here is that $\mathbf{Poly}$ is the full subcategory of $[Set,Set]$ spanned by functors that are accessible and preserve connected limits.

Does this sound right?

Callum Reader (Mar 15 2021 at 23:02):

During the talk I asked a rather vague question about whether or not you could do something `Yoneda-y' with cofunctors. Maybe a better question is: does Cat# have anything like a proarrow equipment structure? Where arrows are cofunctors and I guess maybe proarrows are prafunctors, or some kind of coprofunctor?

David Spivak (Mar 16 2021 at 00:10):

Callum Reader said:

During the talk I asked a rather vague question about whether or not you could do something `Yoneda-y' with cofunctors. Maybe a better question is: does Cat# have anything like a proarrow equipment structure? Where arrows are cofunctors and I guess maybe proarrows are prafunctors, or some kind of coprofunctor?

Hi Callum, there is a proarrow equipment I'm calling \mathbb{P}. It's vertical part is Cat#, and its horizontal part is the parametric right adjoint functors. The category of polynomials in Set, defined by Richard and also David Gepner using those bridge diagrams I <-- E --> B --> J, is the full subequipment spanned by the discrete categories.

Callum Reader (Mar 16 2021 at 11:03):

Oh of course that was the whole point of the talk. My bad. I guess I meant to ask whether or not the bicategory of prafunctors is closed, so you can give the equipment version of the Yoneda lemma.

Daniel Geisler (Mar 16 2021 at 19:32):

(deleted)

Daniel Geisler (Mar 16 2021 at 19:35):

See https://math.stackexchange.com/questions/4061213/golden-exponent-tetration for a current posting about $y^y=y+1$.