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Stream: theory: category theory

Topic: lax reflective subcategories

Jonas Frey (Jan 17 2024 at 02:10):

Has anybody seen this notion?

Call a 2-catgegory inclusion $B\subseteq A$ lax reflective, if it is locally full and full on equivalences, and for every object $a\in A$ there exists an object $\hat a \in B$ and a map $y_a : a\to \hat a$ such that for all $b\in B$, the precomposition map $A(\hat a,b)\to A(a,b)$ has a left adjoint and restricts to an equivalence $B(\hat a, b)\simeq A(a,b)$ (in particular, the inclusion has a left biadjoint, but the biadjunction is not a reflection unless the the subcategory is 2-full).

I think lax reflective subcategories are equivalent to lax idempotent (pseudo)monads on $A$. For every lax idempotent pseudomonad, "being an algebra" and "being an algebra map" are propositions (in the HoTT sense, ie properties), and one can show that the induced subcategory is lax reflective. conversely, it's not hard to see that the monad induced by a lax reflection is lax idempotent.

This seems to give a much simpler definition than that of lax idempotent pseudomonad, since we get basically all of the coherences from universality.

Mike Shulman (Jan 17 2024 at 02:19):

I don't think I've seen that. It sounds interesting.

If you start with a lax-idempotent pseudomonad, how do you deduce that $A(\hat{a},b) \to A(a,b)$ has a left adjoint, rather than just that it's an equivalence when restricted to $B(\hat{a},b)$?

Jonas Frey (Jan 17 2024 at 03:00):

(in the following i'm writing $Ta$ instead of $\hat a$ for the monad)

given an algebra $(b,\beta: Tb\to b)$, the left adjoint $y_!:A(a,b) \to A(\hat a,b)$ is given by
$(f:a\to b)\mapsto\beta\circ Tf$.

then applying the functors back and forth we have
$(f:a\to b)\;\mapsto\;\beta\circ Tf\;\mapsto\;\beta\circ Tf\circ\eta \cong \beta \circ \eta\circ f \cong f$
and in the other direction
$(g:\hat a\to b)\;\mapsto\;g\circ\eta\;\mapsto\;\beta\circ Tg\circ T\eta$
and the counit $(\beta\circ Tg\circ T\eta\to g)$ of the adjunction at $g$ is constructed by the following 'derivation':
$T\eta_a\to\eta_{Ta}$ ('discrete-codiscrete comparison' of the 'adjoint cylinder' $T\eta_a\dashv \mu_a\dashv \eta_{Ta}$)
$Tg\circ T\eta_a\to Tg\circ \eta_{Ta}$ (postcompose with $Tg$)
$Tg\circ T\eta_a\to \eta_b\circ g$ (naturality)
$\beta\circ Tg\circ T\eta_a\to g$ (since $\beta\dashv\eta_b$)

Mike Shulman (Jan 17 2024 at 04:08):

Hmm, interesting. Is there a more abstract way to see that?

Jonas Frey (Jan 17 2024 at 05:28):

I don't know if that's what you mean, but eg the cocontinuous extension of a functor F : C --> X from a small category to a cocomplete category is given by left Kan extension along Yoneda. It seems that this is the general pattern for lax idempotent monads.

Mike Shulman (Jan 17 2024 at 05:32):

I was thinking of a more conceptual proof that doesn't require constructing the unit and counit by hand. But that example does give me a much better intuition for what's going on!

Nathanael Arkor (Jan 17 2024 at 09:24):

I think this may be seen as a combination of two results: (1) that pseudoadjoints may be presented by their action on objects rather than by pseudofunctors (e.g. Lemma 3.7 of FGHW18); (2) that the extension operator of a lax-idempotent pseudomonad is necessarily given by left extension along the unit, which is left-adjoint to precomposition by the unit (this is the presentation of lax-idempotent pseudomonads as "left Kan pseudomonads" in the terminology of MW12).

Nathanael Arkor (Jan 17 2024 at 09:26):

In particular, most of the definition you give is simply the object-assignment presentation of a left pseudoadjoint to the inclusion of $B$ into $A$, which hence defines a pseudomonad. The final condition, that the functor induced by precomposition with the unit has a left adjoint, declares that the pseudomonad induced by the pseudoadjunction is lax-idempotent.

Nathanael Arkor (Jan 17 2024 at 09:28):

It seems like it could be a useful lemma, in any case.

Jonas Frey (Jan 17 2024 at 14:08):

Nathanael Arkor said:

I think this may be seen as a combination of two results: (1) that pseudoadjoints may be presented by their action on objects rather than by pseudofunctors (e.g. Lemma 3.7 of FGHW18); (2) that the extension operator of a lax-idempotent pseudomonad is necessarily given by left extension along the unit, which is left-adjoint to precomposition by the unit (this is the presentation of lax-idempotent pseudomonads as "left Kan pseudomonads" in the terminology of MW12).

Thanks, I was looking for a no-iteration account of lax idempotent monads!

The criterion for the existence of left biadjoints in terms of birepresentability of hom(A,U-) must have been known already before 2018!

Nathanael Arkor (Jan 17 2024 at 14:14):

Jonas Frey said:

The criterion for the existence of left biadjoints in terms of birepresentability of hom(A,U-) must have been known already before 2018!

Indeed. I think it is also Theorem 9.5 of Fiore06, but one would imagine there must be much older references.

Jonas Frey (Jan 17 2024 at 14:22):

Anyway, I've noticed now that lax reflections are not equivalent to lax idempotent monads after all, since the adjunction need not be monadic (for example, the Kleisli category of the cocompletion (downset) monad on Pos is lax reflective in Pos, but not monadic).

The biadjunctions associated to lax reflective subcategories are lax idempotent adjunctions , but not every lax idempotent adjunction corresponds to a lax reflective subcategory, ie the right adjoint need not be full on 2$2-cells and equivalences. This can be seen by noting that for every lax idempotent adjunction with right adjoint $U:B\to A$, there is a lax idempotent adjunction with right adjoint $U\circ\pi_1: B\times(0\to 1)\to B \to A)$.

In conclusion, lax reflections are an intermediate notion between lax idempoent adjunctions and lax idempotent monads.

Jonas Frey (Jan 17 2024 at 14:23):

Nathanael Arkor said:

Jonas Frey said:

The criterion for the existence of left biadjoints in terms of birepresentability of hom(A,U-) must have been known already before 2018!

Indeed. I think it is also Theorem 9.5 of Fiore06, but one would imagine there must be much older references.

Ahh that's a different Fiore than the 2018 paper :smiley:

Mike Shulman (Jan 17 2024 at 16:09):

Jonas Frey said:

In conclusion, lax reflections are an intermediate notion between lax idempoent adjunctions and lax idempotent monads.

Intriguing! Are lax reflective subcategories equivalent to lax-idempotent adjunctions whose right adjoint is locally fully faithful and pseudomonic?

Jonas Frey (Jan 17 2024 at 16:13):

Mike Shulman said:

Jonas Frey said:

In conclusion, lax reflections are an intermediate notion between lax idempoent adjunctions and lax idempotent monads.

Intriguing! Are lax reflective subcategories equivalent to lax-idempotent adjunctions whose right adjoint is locally fully faithful and pseudomonic?

I think so! Unless I'm missing something

Jonas Frey (Jan 17 2024 at 16:15):

And the inclusion of a lax reflective subcategory is monadic whenever the subcategory is closed under "reflective retracts"

Mike Shulman (Jan 17 2024 at 16:58):

In the non-lax case, we have (idempotent monads) = (reflective subcategories) $\subsetneq$ (idempotent adjunctions), the former two being the (necessarily idempotent) adjunctions whose right adjoint is fully faithful. So you're saying that in the lax case, this occurrence of "fully faithful" becomes "locally fully faithful and pseudomonic", and the former equality becomes an inequality also, with equality being equivalent to closure under reflective retracts (noting by comparison that any ordinary-reflective subcategory is closed under all retracts).

Mike Shulman (Jan 17 2024 at 16:59):

That's quite nice!

Jonas Frey (Jan 17 2024 at 20:44):

Mike Shulman said:

In the non-lax case, we have (idempotent monads) = (reflective subcategories) $\subsetneq$ (idempotent adjunctions), the former two being the (necessarily idempotent) adjunctions whose right adjoint is fully faithful. So you're saying that in the lax case, this occurrence of "fully faithful" becomes "locally fully faithful and pseudomonic", and the former equality becomes an inequality also, with equality being equivalent to closure under reflective retracts (noting by comparison that any ordinary-reflective subcategory is closed under all retracts).

Yes, I think that's my understanding!

Jonas Frey (Jan 17 2024 at 20:47):

(but when a pseudo-mono has a left adjoint, the adjunction is not automatically lax idempotent. a counterexample is given by the inclusion of monoids into semigroups, or of rings into non-unital rings)

Mike Shulman (Jan 17 2024 at 20:54):

Right - I guess that difference has to do with the fact that an equivalence can always be improved to an adjoint equivalence, but a non-equivalence adjoint is extra structure.