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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A universally closed morphism is a closed morphism all whose pullbacks are also closed.
Let C be a category with pullbacks and with a notion of closed morphism which is stable under composition and contains all the isomorphisms.
A morphism f:X→Y in C is universally closed if for every h:Z→Y the pullback h *(f):Z× YX→Z is a closed morphism.
In particular, for h=id Y we see that a universally closed morphism is itself closed.