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Stream: learning: questions

Topic: pullback-preserving non-PRA functor

James Deikun (May 09 2025 at 09:32):

Does anyone have an example of a pullback-preserving functor that is not a parametric right adjoint?

James Deikun (May 09 2025 at 13:21):

Actually the inclusion of $\mathbf{FinSet}$ in $\mathbf{Set}$ is one of these, but it doesn't really work as an example for me, so I'm going to amend and ask for an endofunctor.

Mike Shulman (May 09 2025 at 15:51):

Any functor that preserves finite limits but not arbitrary limits should work. Like sheafification for almost any nontrivial Grothendieck topology.

James Deikun (May 09 2025 at 18:13):

Ah, right, a lex reflector is an endofunctor ...

Chaitanya Leena Subramaniam (May 09 2025 at 23:30):

(I'm assuming you mean arbitrary wide pullbacks, not just binary/finite pullbacks --- in which case see Mike's comment)

Any accessible $T:\mathcal C\to\mathcal C$ preserves wide pullbacks iff $T$ admits strict generic factorisations.
(Weber, "Generic morphisms, parametric representations and weakly cartesian monads", Thm 6.6)

If $T:\mathcal C\to\mathcal C$ admits strict generic factorisations, then $T_A : \mathcal C/A\to\mathcal C/TA$ is a right adjoint for every $A$ in $\mathcal C$ (i.e. $T$ is a local right adjoint).
(Weber, "Familial 2-functors and p.r.a.'s", Prop. 2.6)

So a wide pullback-preserving endofunctor $T$ can fail to be p.r.a. in two ways:

$T$ is not accessible---I can't think of an example.
$T$ is accessible, and is a local right adjoint, but $\mathcal C$ does not have a terminal object --- a nice example is the free-polynomial-monad monad on the category PolyEnd of polynomial endofunctors in Set (see J. Kock's "Polynomial functors and trees")

Mike Shulman (May 09 2025 at 23:38):

Inaccessible functors are generally rather tricky to get ahold of. Here is an example of an inaccessible functor between locally presentable categories that preserves all limits (hence a fortiori all wide pullbacks) but is not a right adjoint. It's not an endofunctor, but note that requiring the domain and codomain to be locally presentable also excludes examples like $\rm FinSet \hookrightarrow Set$ .

Chaitanya Leena Subramaniam (May 09 2025 at 23:40):

Nice example!

Mike Shulman (May 09 2025 at 23:50):

An example of an inaccessible endofunctor (of Set, even!) that preserves finite limits (and also finite colimits!) is the ultrapower by the defining ultrafilter on a measurable cardinal (assuming such exists). In fact, Andreas Blass showed in "Exact functors and measurable cardinals" that the existence of a measurable cardinal is equivalent to the existence of a nonidentity endofunctor of Set that preserves finite limits and colimits.