Timeline for Where does Segal's category come from?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 9, 2013 at 22:01 | vote | accept | Qiaochu Yuan | ||
| Oct 9, 2013 at 16:52 | comment | added | Peter May | Qiaochu, it might help to focus more on the point of the definition than on the formalities. If you just use the category of finite sets, you don't have the projections that are used to define the Segal maps. Basepoints give targets to which one can send points (``project coordinates'') one wants to ignore. One wants the domain category to see all of the structure that is automatic in the example n \mapsto X^n. I think that is the key point. | |
| Oct 9, 2013 at 4:10 | comment | added | Qiaochu Yuan | I think what you're saying in the first paragraph amounts to saying that $\text{FinSet}_{\ast}$ is the free "pointed symmetric monoidal category" on a commutative monoid, where by "pointed symmetric monoidal category" I guess I mean a symmetric monoidal category such that the tensor unit is also a zero object. In particular based spaces and so forth are such categories. But again, I don't see why $\text{FinSet}$ hasn't already said everything that needs to be said about basepoints in this story. | |
| Oct 9, 2013 at 4:09 | comment | added | Qiaochu Yuan | So this explanation confuses me because $\text{FinSet}$ already seems to incorporate information about basepoints: namely, the empty set $\emptyset$ is floating around in there already with a unique map to every other object, so if I require that a functor $F : \text{FinSet} \to \text{Top}$ (say) satisfy the property that $F(\emptyset)$ is a point then I already get basepoints on $F(n)$ for all $n$. | |
| Oct 9, 2013 at 2:59 | history | edited | Peter May | CC BY-SA 3.0 |
Small notation fix.
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| Oct 9, 2013 at 2:52 | history | answered | Peter May | CC BY-SA 3.0 |