recently i've been noticing how there are plenty of straightforward examples of finitely cocomplete algebroids that aren't abelian categories, including also symmetric monoidal examples. it now occurs to me that this might mean that (in connection with the "doctrine" ideas that i've been trying to develop) i should switch from techniques involving serre subcategories to a somewhat different flavor of technique...
let's consider for example the category of abelian groups, and what are the full subcategories such that the inclusion has a left adjoint...
we can for example take any homomorphism of abelian groups and force it to become epi. by gabriel-ulmer duality the corresponding full subcategory should consist of those abelian groups such that ...???
let me try again... let's take a sketch of a colimits theory, by which i mean for now that we have a category c and some cocones in c that we're going to declare to be colimit cocones. so the syntactic category of the theory is going to be...
sorry, i'm still undecided about how to proceed here...
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