SQUARE45

A Grothendieck topos is a category

C

which satisfies any one of the following three properties. (A theorem of Jean Giraud states that the properties below are all equivalent.) - There is a small category

D

and an inclusion

C\hookrightarrow \operatorname {Presh} (D)

that admits a finite-limit-preservingleft adjoint. -

C

is the category of sheaves on a Grothendieck site. -

C

satisfies Giraud's axioms, below. Here

\operatorname {Presh} (D)

denotes the category of contravariant functors from

D

to the category of sets; such a contravariant functor is frequently called a presheaf. Giraud's axioms for a category

C

are: -

C

has a small set of generators, and admits all small colimits. Furthermore, fiber products distribute over coproducts; that is, given a set

I

, an

I

-indexed coproduct mapping to

A

, and a morphism

A'\to A

, the pullback is an

I

-indexed coproduct of the pullbacks:

\left(\coprod _{i\in I}B_{i}\right)\times _{A}A'\cong \coprod _{i\in I}(B_{i}\times _{A}A').

- Sums in

C

are disjoint. In other words, the fiber product of

X

and

Y

over their sum is the initial object in

C

. - All equivalence relations in

C

are effective. The last axiom needs the most explanation. If X is an object of C, an "equivalence relation" R on X is a map R → X × X in C such that for any object Y in C, the induced map Hom(Y, R) → Hom(Y, X) × Hom(Y, X) gives an ordinary equivalence relation on the set Hom(Y, X). Since C has colimits we may form the coequalizer of the two maps R → X; call this X/R. The equivalence relation is "effective" if the canonical map

R\to X\times _{X/R}X\,\!

is an isomorphism.

Equivalent definitions

The statement of the theorem