Polynomial functors: Lecture 1/9

Lecture 1/9 in a series on polynomial functors given by David Spivak. These lectures were given as part of the Poly at Work workshop held at the Topos Institute in February 2024.

Topics
0:00 Distributive law
14:26 Universal properties of Sigma and Pi
26:36 Fun(Set, Set)
38:52 Definition of polynomial functors

[MDNQ-udty]

Is there a canonical map that maps polynomial functors in to functors which essentially act the same but without replacement or vice versa?

Poly, when thinking of the representables as "selects" is with replacement. E.g., 5^3 is selecting 3 objects from 5 with replacement which is why we have 5*5*5 obviously. But is there a natural way to convert such expressions into those that are without replacement? 5*4*3? But of course in general. So here 5^3 is interpreted as 3!*5 choose 3. Then one might want to ask if there is a non-commutative form from which we simply get 5^3 is looked as 5 choose 3. My guess is that these are not representable functors and so it actually doesn't work?

Basically algebra typically is in Poly: with replacement but combinatorics is typically without replacement. Is there some transformation between the two?

[randomtiling]

the audio on this series is pretty low, so when I get an ad it kinda blasts my ears