Timeline for Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?
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| Mar 16, 2011 at 0:35 | comment | added | Akhil Mathew | @Qiaochu: thanks for the answer; I'd not thought of $\Delta$ that way. (I have heard, incidentally, that there is a sense in which simplicial sets is the "free model category" on one element -- apparently this is something due to Dwyer, though I don't know enough to make this precise.) | |
| Mar 15, 2011 at 23:58 | comment | added | Harry Gindi | Yes, I realize all that. I put it in scare quotes for a reason =ppp. | |
| Mar 15, 2011 at 15:35 | comment | added | Todd Trimble | @Harry: the point is that you can't properly speak of a monoid in a category unless the monoid is understood relative to a given monoidal structure on that category. Much of your comment didn't quite parse: one is working in the 2-category of monoidal categories, monoidal functors, and monoidal transformations, and it doesn't make much sense to speak of "the universal category containing a monoid", etc. @Qiaochu: Tom Leinster wrote a nice article regarding the whither of $\Delta$ at the Cafe, something like "how I came to love the nerve construction". | |
| Mar 15, 2011 at 15:03 | comment | added | Harry Gindi | @Todd: I was restricting the universal map from the augmented simplex category. | |
| Mar 15, 2011 at 14:40 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Mar 15, 2011 at 14:40 | comment | added | Qiaochu Yuan | @Todd: whoops. Fixing. | |
| Mar 15, 2011 at 14:26 | comment | added | Todd Trimble | Harry, your two sentences are badly in conflict. The augmented simplicial category (the category of finite ordinals and order-preserving maps) is a monoidal category. It is the universal monoidal category equipped with a monoid, in the sense that given any monoidal category equipped with a monoid M, there exists (up to unique monoidal isomorphism) a unique monoidal functor sending the 1-element ordinal (with its unique monoid structure) to M. (I mean, as long as you're correcting someone, do it right!) | |
| Mar 15, 2011 at 12:46 | comment | added | Harry Gindi | The category $\Delta$ is not a monoidal category. It is the universal category "containing a monoid" in the sense that given any category $C$ containing a monoid $M$, there exists a unique functor $\Delta\to C$ sending $[0]$ to $M$ and $[n]$ to the $n+1$-iterated tensor product $M\otimes M\dots \otimes M$, but I'll let you figure out the morphisms =). | |
| Mar 15, 2011 at 10:27 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Mar 15, 2011 at 2:55 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Mar 15, 2011 at 2:49 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Mar 15, 2011 at 2:34 | history | answered | Qiaochu Yuan | CC BY-SA 2.5 |