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Let $C$ be a locally finitely presented category, $A, B$ are two finitely presented (synonym: compact) objects in it. Is it true that $A \times B$ is finitely representable?

At least I have looked at a number of examples of categories of algebras of (finitary) algebraic theories and this seems to be true (although, say, for the category of semigroups the rule for constructing the desired presentation looks unclear). On the other hand, I'm not sure whether this is true for any topos of presheaves?

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    $\begingroup$ In a topos you can define internally finitely presentable objects as those objects $A$ for which the internal hom functor $-^A$ preserves filtered colimits. Then, $-^{A\times B}\cong(-^A)^B$ so product of two internally finitely presentable objects is internally finitely presentable. You can next externalize this using $\hom(A,-)=\Gamma(-^A)$ provided the global elements functor $\Gamma=\hom(1,-)$ preserves filtered colimits. This is related to compactness of the topos itself, that is, it will work if the terminal object $1$ is (externally) finitely presentable. $\endgroup$ Commented May 28, 2023 at 16:30
  • $\begingroup$ @მამუკაჯიბლაძე This is very helpful to me, thanks! $\endgroup$ Commented May 28, 2023 at 16:56

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No. For counterexamples, see Theorems 3.8, 3.9, and 3.10 of

Finiteness properties of direct products of algebraic structures
Peter Mayr, Nik Ruškuc
Journal of Algebra 494 (2018) 167-187.

These theorems show that products of finitely presented loops, idempotent magmas, or lattices may fail to be finitely presentable.

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    $\begingroup$ Thank you very much for the link to the article in which this issue is discussed systematically! $\endgroup$ Commented May 28, 2023 at 16:51

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