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Simonas Tutlys (Sep 28 2022 at 11:25):Greetings,
I know that if we take the limit of a limit diagram we get the same limit, but what if we take the colimit instead? What about a limit of a colimit diagram? Are there any theorems about this? Maybe i missed something.
Morgan Rogers (he/him) (Sep 28 2022 at 11:46):In situations where there is scope to exchange the order in which you take the respective limit and colimit (where the diagram is indexed by a product of categories/graphs, and you're taking the limit over one and the colimit over the other), an important question is whether the two possible orders of the operations produce the same result. More specifically, there is always a "canonical" comparison morphism from the "limit of colimits" to the "colimit of limits", and when this morphism is an isomorphism, we say the limit and colimit "commute". It turns out that, fixing the target category (typically Set) one can characterize classes of diagrams for which this commutation always occurs: "filtered colimits commute with finite limits" is an instance of this type of result.
Simonas Tutlys (Sep 29 2022 at 02:38):Gonna look into commuting (co)limits,that's what i was missing.Thanks!