Timeline for Computation on homotopy colimit cocomplete triangulated categories
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 2, 2020 at 15:10 | vote | accept | user267839 | ||
| Jul 2, 2020 at 15:01 | comment | added | Dmitri Pavlov | @user7391733: Yes, the data itself is insufficient. You cannot define homotopy coherent commutative squares in a triangulated category, for example, because such a square is a quadruple of morphisms f,g,h,k together with a homotopy fg→hk, and there is no way to say what this homotopy is. The point of various enhancements and enrichments is that you move away from bare triangulated categories and add more data to obtain homotopy coherent commutative diagrams. | |
| Jul 2, 2020 at 9:48 | comment | added | user267839 | In simpler analogy: If I want to associate to a certain mathematical object $M$ a group theoretical invariant, then firstly it's meaningful to develop machinery to associate a group $G(M)$ to $M$ in well defined way and then consider the invariants of $G(M)$. Is this the message behind the simplicial enrichment above? | |
| Jul 2, 2020 at 9:47 | comment | added | user267839 | "artificially" endows the involved object with "enough" simplicial structure that then allows to build the homotopy colimit in the described way. Is this the point or did I misunderstood the issue? | |
| Jul 2, 2020 at 9:47 | comment | added | user267839 | Thank you a lot for the answer. One remark on your last paragraph: Do you mean that the message is that if proper coherence (=non triv commut) occures, the data itself not contains not enough intrinsical information to allow to form a homotopy colimit in the way Peter described it? ie that to compute the homotopy colimit as Peter described without enrichment the described object might be not exist/not well defined? That is to make it possible to do it nevertheless, there is a machinery, the simplicial enrichment ncatlab.org/nlab/show/simplicial+object#simplicial_enrichment that | |
| Jul 2, 2020 at 4:01 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |