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For questions on limits and colimts in the sense of category theory, and related notions.

5 votes

Coend computation continued

The step $\int_{a \in A} \mathrm{Set}(\mathrm{hom}_A(a, a), S) = \mathrm{Nat}(\mathrm{hom}_A(-,-),S)$ does not really make sense, because $a \mapsto \mathrm{hom}_A(a,a)$ is not a functor. And $\int^{ …
Reid Barton's user avatar
  • 25.2k
7 votes
Accepted

Commutation of limits and colimits: Is there a choice diagram?

This isn't true in general. Take $I = BG$ and $J = BH$ to be one-object groupoids, so that $A(i, j)$ becomes a set $A$ with commuting actions of $G$ and $H$. The left hand side is obtained by taking t …
Reid Barton's user avatar
  • 25.2k
19 votes

Limits in category theory and analysis

In very special cases, the notions coincide. Let $R$ be the category (poset) whose objects are the real numbers and in which $Hom(x, y)$ has a single element if $x \leq y$ and is empty otherwise. Th …
Reid Barton's user avatar
  • 25.2k