5
$\begingroup$

Let $\mathscr{S}$ be a limit sketch in a small category $\mathcal{E}$, i.e. just a collection of cones in $\mathcal{E}$. Then its category $\mathbf{Mod}(\mathscr{S})$ of models (i.e. functors $\mathcal{E} \to \mathbf{Set}$ which send the cones in $\mathscr{S}$ to limit cones) is cocomplete, in fact locally presentable. It enjoys the following universal property in the $2$-category of cocomplete categories: If $\mathcal{C}$ is any cocomplete category, then $$\mathrm{Hom}_c(\mathbf{Mod}(\mathscr{S}),\mathcal{C}) \simeq \mathbf{Mod}_{\mathcal{C}}(\mathscr{S}^{\mathrm{op}}).$$ Here, $\mathrm{Hom}_c$ denotes the category of cocontinuous functors, $\mathscr{S}^{\mathrm{op}}$ denotes the dual colimit sketch, and $\mathbf{Mod}_{\mathcal{C}}$ refers to $\mathcal{C}$-valued models.

A typical example of this universal property is $\mathrm{Hom}_c(\mathbf{Grp},\mathcal{C}) \simeq \mathbf{CoGrp}(\mathcal{C})$, that is, $\mathbf{Grp}$ is the universal example of a cocomplete category with an internal cogroup object.

The proof is a combination of two well-known results, so I assume that this universal property is also well-known. Can someone confirm this and point me to literature which I can cite for this result?

Maybe it follows from some of the theorems in Kelly's book on enriched categories, chapter 6. But I am not sure.


Here is a sketch of the two-step proof I was thinking of. Let $G : \mathbf{Mod}(\mathscr{S}) \to [\mathcal{E},\mathbf{Set}] $ be the inclusion and $F : [\mathcal{E},\mathbf{Set}] \to \mathbf{Mod}(\mathscr{S})$ its left adjoint. Then by a "tensor-less" variant of Prop. 2.3.6. in Tensor functors between categories of quasi-coherent sheaves (as you see, this is not quite the reference I need!) the category $\mathrm{Hom}_c(\mathbf{Mod}(\mathscr{S}),\mathcal{C})$ is equivalent to the category of those cocontinuous functors $Q^* : [\mathcal{E},\mathbf{Set}] \to \mathcal{C}$ such that $Q^* \to Q^* \circ G \circ F$ is an isomorphism; this is true iff the right adjoint $Q_* : \mathcal{C} \to [\mathcal{E},\mathbf{Set}]$ factors over $\mathbf{Mod}(\mathscr{S})$. By the well-known universal property of $[\mathcal{E},\mathbf{Set}]$ as the free cocompletion of $\mathcal{E}^{\mathrm{op}}$ (for instance, Section 4.4 in Kelly's book), $Q^*$ corresponds to a functor $P : \mathcal{E}^{\mathrm{op}} \to \mathcal{C}$ via $Q^* = P \otimes_{\mathcal{E}} -$, and hence $Q_* = \mathrm{Hom}(P(-),-)$. By the Yoneda Lemma, the requirement that $Q_*$ factors over $\mathbf{Mod}(\mathscr{S})$ exactly means that $P$ is a model of $ \mathscr{S}^{\mathrm{op}}$.


I have found a reference: Theorem 2.2.4 in this paper (actually, I knew this paper, but I somehow forgot that the theorem is in there). But the authors also explicitly state that this theorem is well-known. I would be happy with a more "classical" or "original" reference.

$\endgroup$
1
  • $\begingroup$ The group of Christian Lair and Rene Guitard in Paris 7 Diderot had a sophisticated calculus of sketches that they published in their own private journal called Diagrammes in the 1980s $\endgroup$ Sep 10 at 21:28
5
$\begingroup$

Unfortunately in the form you write, I do not have a one-line-reference, someone might be able to find it. I will provide enough literature and statements in the case in which $\mathscr{S}$ is a small category with finite limits and $\mathbf{Mod}_C(\mathscr{S})$ is defined to be $\mathbf{Lex}(\mathscr{S}, C)$. In the case of a category with finite colimits, the notion of model will be dual and I will choose $\mathbf{Rex}(\mathscr{S}, C)$. This argument can be adapted to the sketch case, but the notation and a couple of details would not look as nice, I will indicate enough literature to accomplish the adaptation at the end of my answer.

Prop. 1 There exists an equivalence of categories, $$ \text{lan}_i(-): \mathbf{Rex}(\mathscr{S}^\circ, C) \leftrightarrows \mathbf{coCont}(\mathbf{Lex}(\mathscr{S}, \text{Set}), C) : i^*.$$

Proof: Let me call $i: \mathscr{S}^\circ \to \mathbf{Mod}(\mathscr{S})$ the Yoneda embedding. This map induces in a natural way a map $i^*: \mathbf{coCont}(\mathbf{Lex}(\mathscr{S}, \text{Set}), C) \to \mathbf{Rex}(\mathscr{S}^\circ, C). $ In the opposite direction we define $\text{lan}_i(-) : \mathbf{Rex}(\mathscr{S}^\circ, C) \to \mathbf{coCont}(\mathbf{Lex}(\mathscr{S}, \text{Set}), C).$ Thus we need to show that,

  1. $i^* \circ \text{lan}_i \cong \mathbb{1}.$
  2. $ \text{lan}_i \circ i^* \cong \mathbb{1}.$

1) follows from Prop. 3.7.3 in Borceux, Handbook of Categorical Algebra, vol. I.

2) follows from Thm. 1.46(v) in Adamek and Rosický, Locally presentable and accessible categories.

If one reads carefully, the whole Prop. 1 follows from LPAC[1.46(v)], my intention was to set enough context to make it more readable.


On the sketch-version.

The sketch version of the statement follows from Prop. 1 via Lemma 4.2.2 in Borceux, Handbook of Categorical Algebra, vol. III. The lemma shows how to replace a (limit) sketch with a finitely complete category without changing its models.

$\endgroup$
5
  • 1
    $\begingroup$ Thank you! It seems that you have a proof by combining two known results, just like me. I was hoping for a reference for the precise statement. Meanwhile I have also found Theorem 6.23 in Kelly's "Basic concepts ...", which is very similar, but not quite the same. (Notice that the $\mathcal{F}$ there is supposed to be a small set, so that it does not include the case of all cocontinuous functors.) $\endgroup$ Jan 4 '20 at 12:47
  • $\begingroup$ @MartinBrandenburg, someone else could do much better than what I have done here. Could you at least mention the two results that you use? I am curious. Also, a direct proof could just work fine. $\endgroup$ Jan 4 '20 at 12:49
  • $\begingroup$ I have added the proof in my question above. $\endgroup$ Jan 30 '20 at 13:04
  • $\begingroup$ Thanks, Martin! $\endgroup$ Jan 30 '20 at 13:05
  • 2
    $\begingroup$ I have found a non-classical reference, see my edit. $\endgroup$ Feb 1 '20 at 10:43
2
$\begingroup$

The case of a Lawvere theory is covered (Theorem 13) in the paper Internal Coalgebras in Cocomplete Categories: Generalizing the Eilenberg-Watts-Theorem, by Porst and Laurent Poinsot. The paper is on arXiv since today.

$\endgroup$
2
$\begingroup$

Probably the earliest reference is Theorem 2.5 in

A. Pultr, The right adjoints into the categories of relational systems, Reports of the Midwest Category Seminar IV. Springer, Berlin, Heidelberg, 1970

Pultr's "relational theories" are exactly small realized limit sketches.

Remark: I have recently generalized the result to large limit sketches (arXiv:2106.11115).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.