Timeline for References for semicategories
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2012 at 22:32 | comment | added | Salvo Tringali | @Ben: I may be missing something, but it seems to me that, while succeeding in the goal of generalizing the notion of "being isomorphic as objects", your idea fails to do the same with the notion of isomorphism, in that it deals only with arrows of type $f: X \to X$ (such that $f^2 = f$). Right? | |
| Sep 12, 2012 at 0:26 | comment | added | Benjamin Steinberg | @Salvo, I don't quite know what to look at there. One thing might be isomorphism of idempotents. If you are familiar with semigroups, then define two idempotents of a semigroupoid to be isomorphic if they are D-equivalent. Observe in a category that two identities are D-equivalent iff the corresponding objects are isomorphic. | |
| Sep 11, 2012 at 22:51 | comment | added | Salvo Tringali | @Todd: You have nothing to be sorry about. Indeed, thank you for your comments. | |
| Sep 11, 2012 at 20:08 | comment | added | Todd Trimble | @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. | |
| Sep 11, 2012 at 18:55 | comment | added | Salvo Tringali | @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... | |
| Sep 11, 2012 at 18:16 | comment | added | Todd Trimble | @Benjamin: no doubt. So the question remains for Salvo: given the likely absence of a systematic treatise, what specifically is wanted? | |
| Sep 11, 2012 at 17:26 | comment | added | Benjamin Steinberg | @Todd, probably nobody could get such a thing published. But the distinction between category and semigroupoid is important because the morphisms are different and adding identities doesn't remedy that. For instance, if one looks at finite monoids, the only ones with no non-trivial homomorphic images are the finite simple groups and the two-elements monoid $(\lbrace 0,1\rbrace,\cdot)$. But there is a whole infinite family of finite semigroups with no non-trivial homomorphisms that are not on this list. An in Morita theory it is useful to look at semifunctors. | |
| Sep 11, 2012 at 16:21 | comment | added | Todd Trimble | Of course, we need to parse it as semigroup-oid, not semi-groupoid! @Salvo: I have a feeling that a "systematic development" on the order of CWM doesn't exist (who would write one? and why?). Can you say what it is more specifically that you want? | |
| Sep 11, 2012 at 14:49 | comment | added | Salvo Tringali | Thank you, Benjamin, for your comments and the reference. I've just checked Tilson's paper: Appendix B indeed deals with semigroupoids but it is three pages long and, as you guessed, not really focused on what I am looking for. | |
| Sep 11, 2012 at 13:56 | comment | added | Benjamin Steinberg | Of course, I am not in favor of the name monoidoid, although logical, for categories! | |
| Sep 11, 2012 at 13:55 | comment | added | Benjamin Steinberg | 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. | |
| Sep 11, 2012 at 13:53 | comment | added | Benjamin Steinberg | Look at the appendix to the paper Categories as Algebra by Bret Tilson, although it may not be what you are looking for. | |
| Sep 11, 2012 at 13:01 | comment | added | Salvo Tringali | 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. | |
| Sep 11, 2012 at 12:23 | comment | added | Jacques Carette | I believe these are also called semigroupoid, for which you'll find more references. | |
| Sep 11, 2012 at 11:37 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
edited body; added 1 characters in body
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| Sep 11, 2012 at 11:26 | history | asked | Salvo Tringali | CC BY-SA 3.0 |