topos theory

category theory

category

functor

(0,1)-topos, Heyting algebra, locale

pretopos

topos

Grothendieck topos

category of presheaves

presheaf

representable presheaf

category of sheaves

site

sieve

coverage, pretopology, topology

sheaf

sheafification

quasitopos

base topos, indexed topos

categorical semantics

internal logic

subobject classifier

natural numbers object

logical morphism

geometric morphism

direct image/inverse image

global sections

geometric embedding

surjective geometric morphism

essential geometric morphism

locally connected geometric morphism

connected geometric morphism

totally connected geometric morphism

étale geometric morphism

open geometric morphism

proper geometric morphism, compact topos

separated geometric morphism, Hausdorff topos

local geometric morphism

bounded geometric morphism

base change

localic geometric morphism

hyperconnected geometric morphism

atomic geometric morphism

topological locale

localic topos

petit topos/gros topos

locally connected topos, connected topos, totally connected topos, strongly connected topos

local topos

cohesive topos

classifying topos

smooth topos

cohomology

homotopy

abelian sheaf cohomology

model structure on simplicial presheaves

higher topos theory

(0,1)-topos

2-topos

2-site

2-sheaf, stack

(∞,1)-topos

(∞,1)-site

(∞,1)-sheaf, ∞-stack, derived stack

Diaconescu's theorem

Barr's theorem

Edit this sidebar

By a “small site” is typically meant a site whose underlying category is a small category. This is the default case, but may be used, for emphasis, in contrast to large site.

But beware that there is also the, unrelated, concept of a little site.