Timeline for Freely adding finite limits preserves some colimits?

5 events

when toggle format	what		by	license	comment
Nov 2, 2017 at 19:37	comment	added	Yonatan Harpaz		@IvanDiLiberti, for a category ${\cal C}$, the Yoneda embedding ${\cal C} \to {\rm Fun}({\cal C}^{{\rm op}},{\rm Set})$ sends $x \in {\cal C}$ to ${\rm Hom}(-,x): {\cal C}^{{\rm op}} \to {\rm Set}$. Now if $x$ is the limit of a diagram $\{x_i\}_{i \in {\cal I}}$ then clearly ${\rm Hom}(-,x) = {\rm lim}_{i \in {\cal I}}{\rm Hom}(-,x_i)$, but if $x$ is the colimit of a diagram then in general there is nothing to be said about maps into $x$. That's why the Yoneda embedding preserves limits and not colimits (or why, in your case, the opposite of Yoneda preserves colimits but not limits).
Oct 30, 2017 at 6:28	comment	added	Mike Shulman		You can't expect to have limits preserved in a free completion under limits, because you're adding new limits freely and in general there's no reason for the new limit of some diagram to coincide with any old limit that it used to have. (There are more refined kinds of limit completion that do preserve certain existing limits, but they aren't "free" cocompletions any more.)
Oct 29, 2017 at 23:02	vote	accept	Ivan Di Liberti
Oct 29, 2017 at 23:02	comment	added	Ivan Di Liberti		Can you add some detail or reference?! For example underling why I have preservation of colimits and not limits?!
Oct 29, 2017 at 20:19	history	answered	Yonatan Harpaz	CC BY-SA 3.0