$\mathbb D$-weighted flatness of functors

Question

It is a matter of unwinding the definitions to see that for a flat (set valued) presheaf the W-weighted colimit functor

W ⊗ _

commutes with finite limits: this is basically because the W-weighted colimit is a colimit over the opposite of the category of elements of W, and since W is flat the latter is filtered. This is in turn equivalent to the request that the Yoneda extension $Yan(W)=Lan_YW$ commutes with finite limits.

This can be generalized to the case of a sound doctrine $\mathbb D$ (see [1]) to get that the following are equivalent for a weight W:

1. Yan(W) is $\mathbb D$-continuous
2. Elts(W)° is $\mathbb D$-filtered
3. W-weighted colimits commute with $\mathbb D$-limits

Let now $\mathbb D$ be a sound doctrine: it is natural to ask whether there are conditions analogous to $\mathbb D$-flatness that ensure that W-weighted colimits commute with U-weighted limits, i.e. that there is an isomorphism

$$W ⊗ \{U, F\} \cong \{U, W⊗F\}$$

for weights W, U and $F : I \times J \to K$ a functor.

This condition stronger than $\mathbb D$-flatness is such that $\mathbb D$-flatness correspond to weighted flatness with respect to conical weights with $\mathbb D$-domain.

Is this condition non-trivial? What is an equivalent condition for weighted $\mathbb D$-flatness?

The only reference I can find for this condition is [2], where no such criterion is stated, at least at a rapid glance.

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[1]: Adámek, Jiří; Borceux, Francis; Lack, Stephen; Rosický, Jiří, A classification of accessible categories, J. Pure Appl. Algebra 175, No.1-3, 7-30 (2002). ZBL1010.18005.

[2]: Lack, Stephen; Rosický, Jiří, Homotopy locally presentable enriched categories, Theory Appl. Categ. 31, 712-754 (2016). ZBL1346.18025.

Answer 1

Dostal and Velebil discuss soundness in the enriched, weighted context. Their main result is

Corollary 3.11. For a locally small saturated class $\Psi$, the following conditions are equivalent:

1. $\Psi$ is sound.
2. $\Psi^+(T^\mathrm{op}) = \Psi\mathrm{-Alg}(T )$ holds for any $\Psi$-theory $T$.
3. For any small $D$, every weight $\varphi: D^\mathrm{op} \to \mathcal{V}$ is a $\Psi$-flat colimit of a diagram in $\Psi(D)$.
4. For any small $D$, the closure of $\Psi(D)$ in $[D^\mathrm{op},\mathcal{V} ]$ under $\Psi$-flat colimits is all of $[D^\mathrm{op},\mathcal{V} ]$.

Let's unpack that. Many of the basic definitions are from Kelly and Schmitt, which in turn is based on Albert and Kelly. But beware that some notation may clash with Kelly and Schmitt.

• $\mathcal{V}$ is a (Bénabou) cosmos in which everything is enriched.

The hypotheses:

• $\Psi$ is a class of weights, which simply means that for every small $\mathcal{V}$-category $D$, we designate some $\Psi(D) \subseteq [D^\mathrm{op},\mathcal{V}]$. We might use the term "weighted limit doctrine" synonymously with "class of weights". We think of $\Psi(D)$ as a "conical" class of weights with respect to which we want to take colimits of functors $D \to C$ (or limits of functors $D^\mathrm{op} \to C$). When $\mathcal{V} = \mathsf{Set}$ and $\mathbb{D}$ is a doctrine in the sense of ABLR, there is a corresponding class of weights $\Psi_\mathbb{D}$ where $\Psi_\mathbb{D}(D)$ consists just of the terminal presheaf if $D \in \mathbb{D}$ and is otherwise empty.

  • $\Psi$ is assumed to be locally small, i.e. $\Psi(D)$ is an essentially small subcategory of $[D^\mathrm{op},\mathcal{V}]$ for every $D$.

  • $\Psi$ is assumed to be saturated (the term used in Kelly and Schmitt for what was called "closed" in Albert and Kelly), i.e. $\Psi(D)$ coincides with the free cocompletion $\Psi^\ast(D)$ of $D$ under colimits of functors $E \to D$ weighted by presheaves in $\Psi(E)$ (recall that $\Psi^\ast(D)$ can be given canonically as a full subcategory of $[D^\mathrm{op},\mathcal{V}]$, so this makes sense). Recall that every class of weights has a saturation $\Psi^\ast$, where we simply replace $\Psi(D)$ by $\Psi^\ast(D)$. The $\Psi$-flat functors and $\Psi^\ast$-flat functors (see below) coincide, so in some sense saturation is a mild condition. But many convenient doctrines (such as finite limits) are not saturated; in order to apply this result we need to think about the saturation of the doctrine and work backwards. Note also that the saturation of a locally small doctrine need not be locally small (e.g. the saturation of the empty class is locally large when $\mathcal{V}$ is the category of suplattices), so there's a bit of synergy between these two conditions.

The key definitions:

• For $\Psi$ a class of weights, Dostal and Velebil define a functor $\varphi: C^\mathrm{op} \to \mathcal{V}$ to be $\Psi$-flat if $\varphi \otimes \{\psi, F\} \to \{\psi, \varphi \otimes F\}$ is an isomorphism for all $D$, all $F: D^\mathrm{op} \otimes C \to \mathcal{V}$ and $\psi \in \Psi(D)$, just as you suggest; this recovers the usual definition of flatness when $\Psi$ is conical.

  • $\Psi^+(D)$ denotes the class of $\Psi$-flat weights.

  • A $\Psi$-theory $T$ is just a $\mathcal{V}$-category with $\Psi$-limits, and $\Psi\mathrm{-Alg}(T)$ is the category of functors $T \to \mathcal{V}$ preserving $\Psi$-limits (think of Lawvere theories or Gabriel-Ulmer duality).

• Let's say that $\varphi$ is representably $\Psi$-flat if the above condition holds for functors $F$ of the form $C(G,c)$ where $G: D \to C$ is a functor. Then Dostal and Velebil's definition of soundness says that $\Psi$ is sound if and only if every representably $\Psi$-flat functor is $\Psi$-flat. This recovers the usual definition when $\Psi$ is conical.

Discussion: Condition (2) says that Gabriel-Ulmer duality holds for the doctrine $\Psi$. So it's a condition that only considers $\Psi$-cocomplete categories. So it's remarkable for it to be equivalent to the other properties. Condition (4) basically says that any colimit is an iterated $\Psi$-flat colimit of $\Psi$-colimits, and Condition (3) says that this can be done in one step.

More can be said for particular doctrines. For example, Dostal and Velebil investigate the doctrine of finite products.