adjoint functor theorem
adjoint lifting theorem
small object argument
Freyd-Mitchell embedding theorem
relation between type theory and category theory
sheaf and topos theory
enriched category theory
higher category theory
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While a category is a called a pointed category if it has a zero object, i.e. if it has an initial object and a terminal object and both are isomorphic, for a quasi-pointed category the last condition is relaxed.
A category is quasi-pointed if it has an initial object 00, a final object 11 and its unique morphism 0→10\to 1 is a monomorphism.