Contents

Definition

Classically, we have:

Since smallness can be relative, we also have:

Note that a Grothendieck topos is a topos because (or if) $S$ is.

The site is not considered part of the structure; different sites may give rise to equivalent category of sheaves.

By the general theory of geometric morphisms, every Grothendieck topos sits inside a category of presheaves by a geometric embedding $\mathrm{Sh}(S)\hookrightarrow \mathrm{PSh}(S)$.

• This may be taken as an alternative definition of sheaf: since Lawvere-Tierney topologies are bijectively given by geometric embeddings, instead of explicitly defining a sheaf as a presheaf satisfying descent, one may define categories of sheaves as geometric embeddings into presheaf categories.

• For details on the relation between the two perspectives see geometric embedding.

• This perspective is useful for defining the vertical categorification of sheaves: stacks and ∞-stacks: the higher categories of these may be defined as geometric embeddings into higher categories of presheaves. This has been worked out in detail for (∞,1)-categories. See (∞,1)-category of (∞,1)-sheaves.

• Sometimes it is useful to distinguish between petit topos and gros topos.

Giraud's axioms

Giraud characterized Grothendieck toposes as categories satisfying certain exactness and small completeness properties (where “small” is again relative to the given category of sets $S$). The exactness properties are elementary (not depending on $S$), and are satisfied in any elementary topos, or even a pretopos.

Giraud’s theorem characterises Grothendieck toposes as follows:

• a locally small category with a small generating set,
• with all finite limits,
• with all small coproducts, which are disjoint, and pullback-stable,
• where all congruences have effective quotient objects, which are also pullback-stable.

These conditions are equivalent to

• a locally small infinitary pretopos with a small generating set.

See Elephant, theorem C.2.2.8.

(See Wikipedia.)

This characterisation may be suitable even when the base category $S$ is only a pretopos, although in that case a Grothendieck ‘topos’ need not actually be a topos. More generally, a Grothendieck topos will inherit properites from $S$, not only having a subobject classifier, but also (for example) having a natural numbers object. Thus the concept makes sense even with weak foundations of mathematics, although it may not have all of the usual properties.

Generalizations

The notion of Grothendieck topos and its characterization from Giraud-type properties can be generalized from the context of categories to that of (∞,1)-categories, where it yields the notion of (∞,1)-topos.

References

A quick introduction of the basic facts of sheaf-topos theory is chapter I, “Background in topos theory” in

• Ieke Moerdijk, Classifying Spaces and Classifying Topoi Lecture Notes in Mathematics 1616, Springer (1995)

A standard textbook on this case is

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

Grothendieck topoi appear around section III,4 there. A proof of Giraud’s theorem is in appendix A.

The proof of Giraud’s theorem for (∞,1)-topoi is section 6.1.5 of

• Jacob Lurie, Higher Topos Theory