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The following is taken from Borceux and Bourn's Mal'cev, Protomodular, Homological, and Semi-Abelian Categories.

Metatheorem 0.1.3. Let $\mathcal P$ be a statement of the form $\varphi\implies \psi$, where $\varphi$ and $\psi$ can be expressed as conjunctions of properties in the following list:

  1. some finite diagram is commutative;
  2. some morphism is a monomorphism;
  3. some morphism is an isomorphism;
  4. some finite diagram is a limit diagram;
  5. an arrow $f:\rightarrow B$ factors (of course, uniquely) through some specified monomorphism $s:S\rightarrowtail B$.

If this statement is valid in the category of sets, it is valid in every category.

After the proof comes the following remark.

It is probably useful to make a comment. The list of properties in our metatheorem is not exhaustive. A better way to express the metatheorem would have been to state it for Horn sentences and the "unique existential quantifier $\exists!$", but we do not want to enter those considerations here.

I tried googling 'Yoneda embedding horn sentence' but didn't find anything I thought was relevant. What is the "better way to express the metatheorem", and why is it true? I'm guessing some topos theory comes in, but I don't really know anything about (the logical aspect of) it.

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  • $\begingroup$ The unstated metatheorem is true because the Yoneda embedding preserves all limits and is fully faithful. Every one of the explicit examples listed can be proved using that fact, and many more besides. $\endgroup$
    – Zhen Lin
    Jun 11, 2016 at 12:43
  • $\begingroup$ @ZhenLin those are the properties I do know, but I don't know how they interact with Horn sentences. Could you briefly explain what Horn sentences are and why the interact well with limit preserving fully faithful functors? $\endgroup$
    – Arrow
    Jun 11, 2016 at 12:50
  • $\begingroup$ Horn sentences are, roughly speaking, a logical way of thinking about finite limits. $\endgroup$
    – Zhen Lin
    Jun 11, 2016 at 13:12
  • $\begingroup$ @ZhenLin could you recommend a light, category-oriented introduction to the topic? $\endgroup$
    – Arrow
    Jun 11, 2016 at 13:17
  • $\begingroup$ Not really. But you can find it in [Sketches of an elephant, Part D]. $\endgroup$
    – Zhen Lin
    Jun 11, 2016 at 13:24

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