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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,


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I believe these are also called semigroupoid, for which you'll find more references. – Jacques Carette Sep 11 '12 at 12:23
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. – Salvo Tringali Sep 11 '12 at 13:01
Look at the appendix to the paper Categories as Algebra by Bret Tilson, although it may not be what you are looking for. – Benjamin Steinberg Sep 11 '12 at 13:53
By the way, until a semigroup is renamed a semi-monoid, I think semigroupoid is a better name than semi-category. There are two good reasons for this. 1) The notion of semigroupoid weakens the axioms of groupoid in the same way that semigroup weakens that of group. 2) Semigroupoid is a multiobject version of a semigroup, rather than a deficient category. – Benjamin Steinberg Sep 11 '12 at 13:55
@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... – Salvo Tringali Sep 11 '12 at 18:55

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