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Stream: theory: category theory

Topic: Examples of non-polynomial comonads

Spencer Breiner (Aug 27 2024 at 14:37):

Polynomial comonads (on Set) are categories, but what about non-polynomial comonads?

I'm trying to understand what these look like, but I don't know any good examples. My go-to non-polynomial functor is an exponential $X\mapsto 2^X$, but this doesn't seem to support a comonad structure. Is there a canonical class of non-polynomial comonads, or good examples that arise naturally?

Morgan Rogers (he/him) (Aug 27 2024 at 14:58):

I guess you mean the covariant powerset functor?

Madeleine Birchfield (Aug 27 2024 at 15:25):

I don't know of any on Set, but for any cohesive topos the flat modality functor $X \mapsto \flat X$ is comonadic but not necessarily a polynomial functor.

(Set is trivially a cohesive topos where the flat modality functor is the identity functor on Set.)

Kevin Carlson (Aug 27 2024 at 16:11):

There are lots of comonads that preserve only finite connected limits (equivalently, pullbacks). The easiest example is the topos of continuous $G$-sets for $G$ a topological group. The cofree functor takes a set $X$ to the cofree not-necessarily-continuous $G$-set on $X$, then to the subset of this on points with open stabilizers, which is the coreflection into continuous $G$-sets. Another example: for any topological space $T$, send a set $X$ to the product of all the skyscraper sheaves on $T$ with stalk $X$, with the product taken over all the points of $T$. This has a left adjoint given by summing a sheaf over all its stalks, and the resulting comonad’s category of coalgebras is exactly the sheaf category. If the sheaf category isn’t of presheaf type then this comonad can’t be polynomial; roughly speaking this happens whenever there are infinitely many points so the stalks don’t preserve the big product taken in the right adjoint.

Kevin Carlson (Aug 27 2024 at 16:14):

In fact a pullback-preserving comonad on Set is exactly the same thing as a Grothendieck topos with a chosen separating set of points, generalizing Ahman-Uustalu’s theorem on poly nomial comonads. There isn’t exactly a reference for this but it’s also not exactly a new result; Garner wrote his ionads paper about such toposes as comonads on slices of Set, and it’s not hard to compose such comonads with the sum-over-fibers comonad to get comonads on Set itself; relatedly, Johnstone has the theorem that the coalgebras for any pullback-preserving comonad on any topos are a topos, generalizing the better-known theorem for left exact comonads.

Morgan Rogers (he/him) (Aug 27 2024 at 16:14):

Kevin Carlson said:

There are lots of comonads that preserve only finite connected limits (equivalently, pullbacks). The easiest example is the topos of continuous $G$-sets for $G$ a topological group. The cofree functor takes a set $X$ to the cofree not-necessarily-continuous $G$-set on $X$, then to the subset of this on points with open stabilizers, which is the coreflection into continuous $G$-sets.

That preserves all finite limits, not just the connected ones (but it is only the finite ones, assuming the topology on $G$ is not closed under infinite intersections).

Kevin Carlson (Aug 27 2024 at 16:15):

Yes, but my next example didn’t :big_smile: the point is that it’s not polynomial.

Kevin Carlson (Aug 27 2024 at 16:17):

You can think of pullback preserving endofunctors of Set, which live in the coproduct completion of the Ind-completion of Set^op, as like polynomials where the directions at each position form a pro-set instead of just a set.

Morgan Rogers (he/him) (Aug 27 2024 at 16:17):

Kevin Carlson said:

Johnstone has the theorem that the coalgebras for any pullback-preserving comonad on any topos are a topos, generalizing the better-known theorem for left exact comonads.

This is definitely the result to know when looking for examples :space_invader:

Kevin Carlson (Aug 27 2024 at 16:18):

I should mention I learned the topological group example from Simon Henry when David Spivak asked the same question last year on MO.

Kevin Carlson (Aug 27 2024 at 16:19):

And I can’t remember for sure, but I think I might still not know any comonads on Set that don’t preserve pullbacks!

Sam Staton (Sep 08 2024 at 06:59):

Presumably cofree comonads for non-pullback-preserving endofunctors on Set will give examples of non-pullback-preserving comonads? For example, start from $P_{\rm{fin}}(A\times -)$, where $P_{\rm{fin}}$ is the finite powerset functor, for which the cofree comonad gives non-well-founded "synchronization trees" modulo bisimilarity.

Kevin Carlson (Sep 08 2024 at 20:01):

That seems plausible! Does the cofree comonad even always exist? The construction I know for the cofree comonad on a polynomial critically uses preservation of connected limits to get convergence in countably many steps. It seems like you know what this particular cofree comonad is, at least, but I’d have trouble figuring it out from the phrase “non well founded synchronization tree”.

Sam Staton (Sep 08 2024 at 21:08):

No! For finitary endofunctors on Set it does, for example On the final sequence of a finitary set functor shows that convergence takes $\omega+\omega$ steps (see also the references in there). That paper goes through the sequence for $P_{\mathrm{fin}}$ in Section 5.