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I am struggling with the notion of structure (for reasons related to Freiman's theory and normed rings), which is the main motivation for my question:

Could you suggest some (good) surveys or books with a systematic development of the theory of semicategories (starting from the basics), in the same spirit, say, of CWM and ACC?

I already checked nLab, and there is not really much about semicategories: The only reference given (click) doesn't provide background on the subject.

Thank you in advance,

Salvo

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  • $\begingroup$ I believe these are also called semigroupoid, for which you'll find more references. $\endgroup$
    – Jacques Carette
    Commented Sep 11, 2012 at 12:23
  • $\begingroup$ Sure, but nonetheless I can't find anything close to my expectations, and this is why I resolved to ask here for advices. E.g., I'd like to know if semicategories have been previously considered by anyone with regard to the definition of a semantics for (finitary) first-order logics. But my concerns are even more primitive, let's say, and this is why I'm looking for a systematic development of the theory from the very basics. $\endgroup$
    – Salvo Tringali
    Commented Sep 11, 2012 at 13:01
  • $\begingroup$ Look at the appendix to the paper Categories as Algebra by Bret Tilson, although it may not be what you are looking for. $\endgroup$
    – Benjamin Steinberg
    Commented Sep 11, 2012 at 13:53
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    $\begingroup$ @Todd. Why not? What is wrong with semicats/semigroupoids? From a philosophical point of view, I've always believed - were it nothing but my silly faith - that minimalism, which is the name that I use to refer to the absence of conceptual redundancies, is the right way to go to shed light on sometimes obscure aspects of our most shining theories (and biases), in and out of the mathematical reality. In any case, you asked for a specific example. So, here it is: What should it be an isomorphism in the setting of semicats, as we cannot count on local identities? I've got my own idea, but... $\endgroup$
    – Salvo Tringali
    Commented Sep 11, 2012 at 18:55
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    $\begingroup$ @Salvo: Sorry, I didn't mean to give the impression they weren't worth studying; it's just that I'm not aware of any current research on these things, and therefore I didn't think you were going to have any luck locating a systematic treatise devoted to them. Failing that, it makes sense to ask a more focused question (which you did). I don't have any ideas on this myself, but Benjamin is highly knowledgeable about semigroups, I believe, and hopefully he can help. $\endgroup$
    – Todd Trimble
    Commented Sep 11, 2012 at 20:08

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