category theory

category

functor

natural transformation

Cat

universal construction

representable functor

adjoint functor

limit/colimit

weighted limit

end/coend

Kan extension

Yoneda lemma

Isbell duality

Grothendieck construction

adjoint functor theorem

monadicity theorem

adjoint lifting theorem

Tannaka duality

Gabriel-Ulmer duality

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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The empty category is the category with no objects (and hence no morphisms). It is a groupoid, so we may call it the empty groupoid. One can similarly speak of the empty $n$-category, the empty $\infty$-groupoid, etc etc etc.

The empty category is discrete, hence may be identified with a set: the empty set. This set is a subsingleton, so we may also identify it with a truth value: the false one.

The empty category is initial in Cat (as well as Grpd, ∞Grpd, Set, etc).