local topos, connected topos, cohesive topos,
local (∞,1)-topos, ∞-connected (∞,1)-topos, cohesive (∞,1)-topos
Edit this sidebar
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 codiscrete objects if it has a full and faithful right adjoint coDisc:ℬ↪ℰ.
This is for instance the case for the global section geometric morphism of a local topos (Disc⊣Γ⊣coDisc)ℰ→ℬ.
In this situation, we say that a concrete object X∈ℰ is one for which the (Γ⊣coDisc)-unit of an adjunction is a monomorphism.
If ℰ is a sheaf topos, this is called a concrete sheaf.
If ℰ is a cohesive (∞,1)-topos then this is called a concrete (∞,1)-sheaf or the like.
The dual notion is that of a co-concrete object.
Γ is a faithful functor on morphisms whose codomain is concrete.
(shape modality ⊣ flat modality ⊣ sharp modality)
discrete object, codiscrete object, concrete object
structures in cohesion
(reduction modality ⊣ infinitesimal shape modality ⊣ infinitesimal flat modality)
(Red⊣ʃ inf⊣♭ inf)
reduced object, coreduced object, formally smooth object
formally étale map
structures in differential cohesion