Timeline for Duality in Spc with ∧ and [-,-]
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 15 at 10:54 | comment | added | Marc Hoyois | Homotopy colimit constructions are not defined in the homotopy category, that's why this only makes sense in the $\infty$-category. Even then, the correct definition of $\wedge_A$ is as the colimit of a whole simplicial diagram (the so-called bar construction), not just as a coequalizer. | |
| Feb 14 at 13:18 | comment | added | user30211 | @MarcHoyois do you mean that they aren't as extensive as E₁-monoids and so less natural, or rather that there is no definition? Because it seems like you can have taken a homotopy coequilizer of the two maps X ∧ A ∧ Y ⭢ X ∧Y, and a homotopy equalizer of the two maps A ∧ [X,Y] ⭢ [X,Y]. | |
| Feb 11 at 19:09 | comment | added | Marc Hoyois | Actions in the homotopy category are too incoherent to define $-\wedge_A-$ and $[-,-]_A$. These constructions only make sense for actions of $E_1$-monoids in the $\infty$-category of spaces (they are defined by a simplicial colimit and cosimplicial limit, respectively), and $A$ should be at least $E_2$ if you want the output to still have an action of $A$. | |
| Feb 11 at 7:15 | history | asked | user30211 | CC BY-SA 4.0 |