Timeline for Universal cocompletion without leaving our universe
Current License: CC BY-SA 2.5
3 events
| when toggle format | what | by | license | comment | |
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| Feb 18, 2011 at 17:33 | comment | added | Todd Trimble | @Peter: yes, you seem to have gotten those sizes right. You probably know this already (hence this might be a nitpick), but I wouldn't write $\hom(-^{op}, \mathcal{U})$: it's okay on objects, but as written that's contravariant on 1-cells. The small-cocompletion monad $P$ applied to a functor $f: C \to D$ between $\mathcal{U}$-small categories is the left adjoint to $\hom(f^{op}, \mathcal{U})$, in other words is the left Kan extension along $f^{op}$. | |
| Feb 18, 2011 at 16:23 | comment | added | Peter LeFanu Lumsdaine | $\newcommand{\C}{\mathbf{C}} \newcommand{\U}{\mathcal{U}} \newcommand{\Hom}{\mathrm{Hom}} \newcommand{\op}{\mathrm{op}}$…and a very nice answer! Quick check on what smallness conditions are where: Am I right in understanding this as an adjoint from “$\U^+$-small, $\U$-locally-small categories” to “$\U^+$-small, $\U$-locally-small, $\U$-cocomplete categories”? And when restricted to “$\U$-small categories”, it does agree, up to (?pseudo-)natural equivalence, with the presheaves construction $\Hom(-^\op,\U)$? | |
| Feb 18, 2011 at 15:32 | history | answered | Todd Trimble | CC BY-SA 2.5 |