Timeline for Projective objects in HTT
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 7, 2014 at 19:06 | vote | accept | fosco | ||
| Nov 11, 2013 at 1:02 | comment | added | Zhen Lin | @tetrapharmakon What Jacob Lurie has described is the $\infty$-analogue of the observation that the ordinary colimit for a diagram $\mathbf{\Delta}^\mathrm{op} \to \mathcal{C}$ is the same thing as the colimit for the same diagram weighted by the terminal presheaf on $\mathbf{\Delta}^\mathrm{op}$ (a.k.a. "functor tensor product"). The fact that geometric realisation (and more generally left Kan extension) can be expressed in terms of weighted colimits is not conceptually helpful here, in my view. | |
| Nov 10, 2013 at 22:38 | comment | added | fosco | @JacobLurie I have to meditate more on this (especially because Zhen Lin and you seem to disagree). In the meanwhile, thank you again! | |
| Nov 6, 2013 at 18:25 | comment | added | Jacob Lurie | What you describe works exactly the same way in the setting of quasi-categories: if C is a category which admits small colimits, then any cosimplicial object of C determines a pair of adjoint functors relating C to the quasicategory of simplicial spaces. If C is the quasi-category of spaces and your cosimplicial space is contractible in each degree, the resulting functor from simplicial spaces to spaces is given by taking the colimit. | |
| Nov 6, 2013 at 18:00 | comment | added | Zhen Lin | @tetrapharmakon This notion is entirely different. Rather it is based on the observation that the geometric realisation of a simplicial space is (homotopy equivalent to) its homotopy colimit. | |
| Nov 6, 2013 at 16:57 | comment | added | fosco | Thank you for your answer. My problem is that I would like to establish a link between the two definitions. To my eye, geometric realizations arise as Yoneda extension of certain dense functors $\Delta\to \bf C$ to a cocomplete category (cocompleteness implies this Kan extension has a right adjoint, density of $\iota$ that this right adjoint is a fully faithful embedding; this is well-known). Is it possible to generalize this pattern to the quasicategorical setting? If yes, is there a link with the definition of projective object you (& Joyal) gave? | |
| Nov 6, 2013 at 11:54 | history | answered | Jacob Lurie | CC BY-SA 3.0 |