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I was wondering if anyone could add some more examples to the pages on powers and copowers?
The only examples featured on both pages are the obvious ones:
a closed monoidal category is (co)powered over itself
(co)complete locally small categories are (co)powered over Sets.
For instance, are there any good examples of abelian categories, such as abelian sheaves, which are (co)powered over abelian groups? Or categories which are (co)powered over chain complexes?
This was asked also at MO; I guess you are the author. I think the reason the question wasn’t received better is that there’s an “embarrassment of riches”: most of the reasonable examples encountered in the wild, such as abelian sheaves, have powers and copowers. I added a comment on that at MO.
Here’s an example where an enriched category is $Set$-complete but not complete in the enriched sense: consider monoids $M$ as 1-object categories $B M$, so that monoids are enriched in $Cat$ (the $Cat$-hom being given by $Cat(B M, B N)$). Then $Cat$-powers don’t exist in $Mon$. For example, if $\mathbf{2}$ is the arrow category and $M$ has more than one element, then the power $M^\mathbf{2}$ doesn’t exist in $Mon$ (in $Cat$ it’s the category whose objects are elements of $M$ and where morphisms $m \to n$ are elements $a$ such that $a m = n a$).
Thanks! Your answer on MO was very helpful. When I have some time I’ll add some of your examples to the nLab page.
Thanks, Bartek. I had added something ever so brief to copower, but there’s room for other additions. :-)
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