Timeline for topological monoid from symmetric monoidal category
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Mar 23, 2021 at 15:15 | history | edited | YCor |
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| Dec 4, 2012 at 15:56 | vote | accept | Ulrich Pennig | ||
| Dec 4, 2012 at 14:41 | answer | added | Peter May | timeline score: 5 | |
| Dec 4, 2012 at 6:54 | vote | accept | Ulrich Pennig | ||
| Dec 4, 2012 at 15:56 | |||||
| Dec 3, 2012 at 19:35 | comment | added | Ulrich Pennig | @Peter May: You are right. I tried to clarify the question to prevent people from having false hopes about too much commutativity. | |
| Dec 3, 2012 at 19:29 | answer | added | Ulrich Pennig | timeline score: 3 | |
| Dec 3, 2012 at 19:26 | history | edited | Ulrich Pennig | CC BY-SA 3.0 |
deleted symmetric
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| Dec 2, 2012 at 23:47 | comment | added | Peter May | You not only can but should drop the word ``symmetric''. Otherwise someone in a naive audience may ask whether your topological monoid is commutative, and of course it is not. With the standard notion of a strict symmetric monoidal category (aka a permutative category), the classifying space gives rise to a spectrum whose zeroth space is a group completion of your monoid. | |
| Nov 28, 2012 at 1:08 | comment | added | Todd Trimble | @Theo: I didn't think you were being impolite; I just thought you were seeking clarification (and since I was suggesting it, I thought I'd better answer!). | |
| Nov 28, 2012 at 0:25 | comment | added | Martin Brandenburg | As Todd already pointed out, this is special case of the observation that every lax monoidal functor $C \to D$ extends to a functor $\mathsf{Mon}(C) \to \mathsf{Mon}(D)$ (applied to $C=(\mathsf{Cat},\times)$ and $D=(\mathsf{CGHaus},\times)$). Since this is trivial, it is always just mentioned in the literature (for example Saavedra Rivano, Categories Tannakiennes, I.6.1.4.). | |
| Nov 27, 2012 at 23:46 | comment | added | Theo Buehler | @Todd: Apologies for the unfortunate formulation of my objection to this "standard manever" -- I didn't mean to be impolite. It certainly is standard in certain circles, but not so much in others, and it is sometimes hard to tell for outsiders like me. I for one fell into the trap of believing that the other concept was intended more than once... @Ulrich: sorry for the noise. | |
| Nov 27, 2012 at 23:33 | comment | added | Ulrich Pennig | @Theo: sorry for being so vague with the statement in the question. | |
| Nov 27, 2012 at 23:12 | comment | added | Todd Trimble | Theo, if you're asking Todd, then he'd say "topological monoid" here is an abuse of language where strictly speaking we are taking $kSpace$ as our "convenient" category of topological spaces. I thought that was a pretty standard maneuver; it's well known that $Top$ has some undesirable properties (such as not being cartesian closed, when we'd really like function spaces with all our hearts). | |
| Nov 27, 2012 at 22:39 | comment | added | Theo Buehler | How does Todd's suggestion give a topological monoid without further assumptions like countability or local finiteness? The forgetful functor $\mathsf{kSpace \to Top}$ does not preserve products, so what you get is at best something with a multiplication which is continuous on compact subsets of the product in $\mathsf{Top}$. | |
| Nov 27, 2012 at 19:49 | comment | added | Ulrich Pennig | @Todd: Thank you. I will have a look. I know that the proof of this is really just the fact that geometric realizations are product preserving. But I happen to write a paper, which uses this and has a main audience outside of topology. So I need to back things up with as many references as possible. | |
| Nov 27, 2012 at 19:26 | comment | added | Todd Trimble | Of course, you can drop the word "symmetric": a strict monoidal category is the same as a monoid in $(Cat, \times)$, and since the classifying space functor $Cat \to kSpace$ preserves products, it takes monoids to monoids. This isn't covered in Segal's Categories and Cohomology Theories? (Maybe it is; I don't recall.) | |
| Nov 27, 2012 at 19:00 | history | asked | Ulrich Pennig | CC BY-SA 3.0 |