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Grothendieck topology

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Idea

A Grothendieck topology on a category is a choice of morphisms in that category which are regarded as covers.

A category equipped with a Grothendieck topology is a site. Sometimes all sites are required to be small.

Probably the main point of having a site is so that one can define sheaves, or more generally stacks, on it. In particular, the category of sheaves on a (small) site is a Grothendieck topos.

Definition

Definition

A Grothendieck topology JJ on a category CC is an assignment to each object cCc \in C of a collection of sieves on cc which are called covering sieves, satisfying the following axioms:

  1. If FF is a sieve that covers cc and g:dcg: d \to c is any morphism, then the pullback sieve g *Fg^* F covers dd.

  2. The maximal sieve id:hom(,c)hom(,c)id: \hom(-, c) \hookrightarrow \hom(-, c) is always a covering sieve;

  3. Two sieves F,GF, G of cc cover cc if and only if their intersection FGF \cap G covers cc. (Here the saturation condition is important.)

  4. If FF is a sieve on cc such that the sieve d{g:dcg *Fcoversd}\bigcup_d \{g: d \to c| g^* F \; covers \; d\} is a covering sieve of cc, then FF itself covers cc.

The set of covering sieves of an object cc is denoted J(c)J(c).

A category equipped with a Grothendieck topology is called a site .

Remark

The first axiom guarantees that we have a functor J:C opSetJ: C^{op} \to Set. Thus JJ itself can be regarded as an object of the presheaf topos [C op,Set][C^{op},Set]; in this way Grothendieck topologies on CC are identified with Lawvere-Tierney topologies on [C op,Set][C^{op},Set].

Given a Grothendieck topology JJ on a small category CC, one can define the category Sh(C,J)Sh(C,J) of sheaves on CC relative to JJ, which is a reflective subcategory of the category [C op,Set][C^{op},Set] of presheaves on CC. Thus we have a functor CSh(C,J)C\to Sh(C,J) given by the composite of the Yoneda embedding with the reflection (or “sheafification”). This composite functor is fully faithful if and only if all representable presheaves are sheaves for JJ; a topology with this property is called subcanonical.

Saturation

Grothendieck topologies may be and in practice quite often are obtained as closures of collections of morphisms that are not yet closed under the operations above (that are not yet sieves, not yet pullback stable, etc.).

Two notions of such unsaturated collections of morphisms inducing Grothendieck topologies are

Examples

Topology on open subsets of a topological space

The archetypical example of a Grothendieck topology is that on a category of open subsets Op(X)Op(X) of a topological space XX. A covering family of an open subset UXU \subset X is a collection of open subsets V iUV_i \subset U that cover UU in the ordinary sense of the word, i.e. which are such that every point xUx \in U is in at least one of the V iV_i.

Regular topology

Any regular category CC admits a subcanonical Grothendieck topology whose covering families are generated by single regular epimorphisms. If CC is exact or has pullback-stable reflexive coequalizers, then its codomain fibration is a stack for this topology (the necessary and sufficient condition is that any pullback of a kernel pair is again a kernel pair).

Extensive topology

Any extensive category admits a Grothendieck topology whose covering families are (generated by) the families of inclusions into a coproduct (finite or small, as appropriate). We call this the extensive coverage or extensive topology. The codomain fibration of any extensive category is a stack for its extensive topology.

Coherent topology

Any coherent category CC admits a subcanonical Grothendieck topology in which the covering families are generated by finite, jointly regular-epimorphic families. Equivalently, they are generated by single regular epimorphisms and by finite unions of subobjects. If CC is extensive, then its coherent topology is generated by the regular topology together with the extensive topology. (In fact, the coherent topology is superextensive.)

Canonical topology

On any category there is a largest subcanonical topology. This is called the canonical topology, with “subcanonical” a back-formation from this (since a topology is subcanonical iff it is contained in the canonical topology). On a Grothendieck topos, the covering families in the canonical topology are those which are jointly epimorphic.

A more general notion is simply a collection of “covering families,” not necessarily sieves, satisfying only pullback-stability; this suffices to define an equivalent notion of sheaf. Following the Elephant, we call such a system a coverage. A Grothendieck topology may then be defined as a coverage that consists of sieves (which the Elephant calls “sifted”) and satisfies certain extra saturation conditions; see coverage for details.

An intermediate notion is that of a Grothendieck pretopology, which consists of covering families that satisfy some, but not all, of the closure conditions for a Grothendieck topology. Many examples are “naturally” pretopologies, but must be “saturated” under the remaining closure conditions to produce Grothendieck topologies.

As remarked above, Grothendieck topologies on a small category CC are also in bijective correspondence with Lawvere-Tierney topologies on the presheaf topos [C op,Set][C^{op},Set]. See Lawvere-Tierney topology for a description of the correspondence.

See also

References

Standard texbooks inlcude

Discussions of variants of the notion and its variants is at historical notes on Grothendieck topology.

Revised on February 16, 2012 02:09:51 by Urs Schreiber (82.169.65.155)