local topos, connected topos, cohesive topos,
local (∞,1)-topos, ∞-connected (∞,1)-topos, cohesive (∞,1)-topos
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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
For Γ:ℰ→ℬ a functor we say that it has discrete objects if it has a full and faithful left adjoint Disc:ℬ↪ℰ.
An object in the essential image of Disc is called a discrete object.
This is for instance the case for the global section geometric morphism of a connected topos (Disc⊣Γ):ℰ→ℬ.
If one thinks of ℰ as a category of spaces, then the discrete objects are called discrete spaces.
The dual notion is that of codiscrete objects.