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In Accessible categories : the foundations of categorical model theory, chapter 3 p.58, Makkai and Paré claim that there is "an (obvious) identification of a class of sketches so that the categories Mod(S) for such sketches S are precisely the categories of models of complete theories with elementary embeddings as morphisms".

In other terms, there seems to be a "sketchable" counterpart of the property of completeness of a formal theory. But there is no explicit reference in the book.

What can be this sketch-theoretical property for such identification? Is there any paper where this identification is explicit?

Many thanks, in advance.

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1 Answer 1

This isn't a direct answer, but rather consists of two references (in the same book) which when you put them together provides a proof that all elementary categories are sketchable.

The book is "Locally Presentably Categories and Accessible Categories" by Adamek and Rosicky.

The first reference, Theorem 5.42 and Theorem 5.44. These theorems together show that a category is accessible if and only if it is equivalent to a cateory whose objects are models of some theory $T \subseteq L_{\kappa,\kappa}$ and whose morphisms consist of all embeeddings which preserve all formulas of $L_{\kappa, \kappa}$.

In particular this implies that for any first order theory $T$ the category whose objects are of models of $T$ and whose morphisms are elementary embeddings is accessible.

The second reference is Chapter 2F, and in particular Theorem 2.58 and Theorem 2.60 where they show a category is accessible if and only if it is sketchable.

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