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Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
4
votes
Accepted
Bi-embeddability vs. isomorphism
I don't know if this counts as a "naturally-occuring" algebraic category but as I scanned through my internal list of categories of modules over a ring, I spotted that the following has (1) and (2).
…
2
votes
How to construct pair of adjoint functors from category A to category A_D(category of diagrams)
There are several pairs of adjoint functors of the kind you desire but it isn't clear to me if any (or all) of them will give you enough projectives in $A^D$.
For example for each $d$ in $D$ $\iota_d …