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There are multiple ways to formalize the notion of a (limit) sketch, which are basically equivalent. This makes it a bit difficult to decide on a "right way" to formalize sketches. One nice property would be that a category of models (in say $\mathsf{Set}$) is given by a unique theory up to equivalence.

My (probably flawed) understanding is that based on a particular weak definition of "sketch" this need to be the case and that a sketch may care about the way we axiomatize our theory, like e.g. whether we use a constant and a binary operation to say what a monoid is or a sequence of operators $(\prod : A^n \to A)$ or something else.

Let's only focus on limit sketches (since these yield the theories I actually care about).

What is the correct notion of limit sketch where category of models are given by a unique theory up to equivalence?

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  • $\begingroup$ Do you know the notion of a limit sketch in Adamek-Rosicky's book on locally presentable and accessible categories? It seems quite natural to me. But I don't know if this answers your question (which I don't fully understand, tbh). $\endgroup$
    – HeinrichD
    Commented Oct 14, 2016 at 16:57
  • $\begingroup$ @HeinrichD Consider some algebraic theory. There are multiple syntactical theories, i.e. pairs of the form (signature, equations) that are equivalent in the sense that they give you the same category of models in $\mathsf{Set}$. Now there should be some method for turning a syntactical theory into a limit sketch with the same models in $\mathsf{Set}$. I'm unsure whether the limit sketches for different but equivalent (same models) syntactical theories are actually equivalent as categories or not, e.g. using the definition by Adamek-Rosicky. That would be a desirable property to have. $\endgroup$
    – Stefan Perko
    Commented Oct 14, 2016 at 17:37
  • $\begingroup$ Or to be more blunt: I want $\operatorname{Mod} \mathcal{T} \simeq \operatorname{Mod} \mathcal{T'} \Rightarrow \mathcal{T} \simeq \mathcal{T'}$ (if the category of models are the same then so are the sketches). $\endgroup$
    – Stefan Perko
    Commented Oct 14, 2016 at 17:40
  • $\begingroup$ @StefanPerko that is a really strong requirement... never the less I would say that the answer is true if both $\mathcal T$ and $\mathcal T'$ have the same signature. Are you looking for something of more general? $\endgroup$
    – Giorgio Mossa
    Commented Oct 14, 2016 at 17:49
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    $\begingroup$ Yes, indeed; that's a good paper. And have you looked at Adamek and Rosicky's book? There are extensions of Gabriel-Ulmer duality there as well. I regard (limit) sketches as akin to signatures for algebraic theories: they are good for concise presentations, but in order to get a more invariant notion of theory, you have to "saturate" the class of (partial) operations; in the classical Gabriel-Ulmer case this is done by passing to the category of finitary right adjoints $\mathcal{M} \to \text{Set}$ where $\mathcal{M}$ is the given category of models. $\endgroup$
    – Todd Trimble
    Commented Oct 15, 2016 at 11:51

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If I understand correctly your question you are looking for some definition of limit-sketches such that if two sketches $\mathcal T$ and $\mathcal T'$ are Morita-equivalent (that is the categories of their $\mathbf{Set}$-valued models are equivalent) then they are equivalent as categories.

If that is the case they you are doomed to fail. The problem lies down to the fact that any reasonable definition of limit-sketch should comprise algebraic theories as a special case, meaning that any algebraic (i.e. Lawvere) theory should be a limit-sketch.

Unfortunately it is well know that there are non equivalent algebraic theories giving rise to the same categories of algebras (models).

As an example, taken from Adàmek, Rosicky and Vitale's Algebraic theory, you can consider the algebraic theories $\mathcal N$, the full sub-category of $\mathbf{Set}$ spanned by the finite sets $[n]=\{1,\dots,n\}$, and its full subcategory $\mathcal T_2$ spanned by sets of the form $[2n]$. In the above mentioned book is proven that the category of models $\text{Alg}(\mathcal N)$ and $\text{Alg}(\mathcal T_2)$ are both equivalent to $\mathbf{Set}$ but that two theories are not equivalent as categories.

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  • $\begingroup$ Ah... I was completely unaware of this. One thing though: The book (that I actually had already...) mentions that two Lawvere theories are Morita equivalent iff they have equivalent idempotent completions. Do you know whether the idempotent completion of a limit sketch can be used as a theory in place of the limit sketch itself somehow? $\endgroup$
    – Stefan Perko
    Commented Oct 14, 2016 at 21:00
  • $\begingroup$ Unfortunately I am not aware of that. On the other hand I am not even sure if a similar result extend to limit-sketches. $\endgroup$
    – Giorgio Mossa
    Commented Oct 14, 2016 at 21:34

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