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.
In the original definition (Michael Artin‘s seminar notes “Grothendieck topologies”), a Grothendieck topology on a category $C$ is defined as a set $T$ of coverings satisfying certain closure properties.
More precisely (to cite from Artin’s notes):
A Grothendieck topology on a category $C$ is a set $T$ of families of maps $\{\phi_i\colon U_i\to U\}_{i\in I}$ (known as coverings) such that
for any isomorphism $\phi$ we have $\{\phi\}\in T$;
if $\{U_i\to U\}\in T$ and $\{V_{i,j}\to U_i\}\in T$ for each $i$, then $\{V_{i,j}\to U\}\in T$;
if $\{U_i\to U\}\in T$ and $V\to U$ is a morphism, then $U_i\times_U V$ exist and $\{U_i\times_U V\to V\}\in T$.
This is almost identical to the current definition of Grothendieck pretopology, except that in Artin’s definition only the relevant pullbacks are required to exist.
If $g:d\to c$ is a morphism in a category $C$ and $F\subset C(-,c)$ a sieve on $c$ then
is a sieve on $d$, the pullback sieve of $F$ along $g$.
The following definition can be found in SGA 4, Exposé II.
A Grothendieck topology $J$ on a category $C$ is an assignment to each object $c \in C$ of a collection of sieves on $c$ which are called covering sieves, satisfying the following axioms:
(Stability under base change.) If $F$ is a sieve that covers $c$ and $g: d \to c$ is any morphism, then the pullback sieve $g^* F$ covers $d$.
(Local character condition.) If $F$ is a sieve on $c$ such that the sieve $\bigcup_d \{g: d \to c| g^* F \; covers \; d\}$ contains a covering sieve of $c$, then $F$ itself covers $c$.
The maximal sieve $id: \hom(-, c) \hookrightarrow \hom(-, c)$ is always a covering sieve;
The set of covering sieves of an object $c$ is denoted $J(c)$.
A category equipped with a Grothendieck topology is called a site .
The axiom of stability under base change guarantees that we have a functor $J: C^{op} \to Set$. Thus $J$ itself can be regarded as an object of the presheaf topos $[C^{op},Set]$; in this way Grothendieck topologies on $C$ are identified with Lawvere-Tierney topologies on $[C^{op},Set]$.
Given a Grothendieck topology $J$ on a small category $C$, one can define the category $Sh(C,J)$ of sheaves on $C$ relative to $J$, which is a reflective subcategory of the category $[C^{op},Set]$ of presheaves on $C$. Thus we have a functor $C\to Sh(C,J)$ 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 $J$; a topology with this property is called subcanonical.
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
The archetypical example of a Grothendieck topology is that on a category of open subsets $Op(X)$ of a topological space $X$. A covering family of an open subset $U \subset X$ is a collection of open subsets $V_i \subset U$ that cover $U$ in the ordinary sense of the word, i.e. which are such that every point $x \in U$ is in at least one of the $V_i$.
Any regular category $C$ admits a subcanonical Grothendieck topology whose covering families are generated by single regular epimorphisms. If $C$ 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).
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.
Any coherent category $C$ 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 $C$ is extensive, then its coherent topology is generated by the regular topology together with the extensive topology. (In fact, the coherent topology is superextensive.)
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.
The term “topology” for this concept is hallowed by tradition and the name of Grothendieck, but some people consider it to be somewhat unfortunate. While “a category equipped with a (Grothendieck) topology” is, in fact, sort of a generalization of “a set equipped with an (ordinary) topology,” the relationship between a category and its topology is quite different from the relationship between a set and its topology. In particular, when we construct a site from a topological space, the objects of the category are the open sets, not the points, of the space.
This use of “topology” can also lead to confusion since for a topologist, there is a completely different and very natural meaning of “topology on a category:” namely, topologies on its sets of objects and morphisms making it into an internal category in Top. The topologist’s definition is, of course, a conservative extension of the classical notions of “topology on a set” and even “topology on a group,” while there are no nontrivial Grothendieck topologies on a group considered as a 1-object category.
Furthermore, the use of “topology” for a category with a system of covers also leads to the use of “continuous” for a functor which preserves covers. This is (in some people’s opinion) doubly unfortunate, since “continuous functor” is used not only by topologists to mean an internal or enriched functor over Top, but also by many category-theorists to mean a functor that preserves all small limits. This is especially confusing since covers are more akin to colimits than limits. Moreover, while a continuous function between topological spaces does induce a a cover-preserving functor between categories of open sets, the function and the functor go in opposite directions.
In Sketches of an Elephant, Peter Johnstone introduced the term coverage used above for a system of covers on a category, with “Grothendieck coverage” as a proposed replacement for “Grothendieck topology.” See coverage for his definitions. Proposed replacements for “Lawvere-Tierney topology” include:
Local operator (used by Johnstone).
Local modality or geometric modality, since in the internal logic of the topos, it represents a modal operator with the intutive meaning of “it is locally the case that…”.
Lawvere-Tierney operator or Lawvere-Tierney modality to avoid possible conflict with other uses of “local” or “geometric” in modal logic.
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 $C$ are also in bijective correspondence with Lawvere-Tierney topologies on the presheaf topos $[C^{op},Set]$. See Lawvere-Tierney topology for a description of the correspondence.
See also
The original definition is in
The definition in terms of sieves is found in
Standard books include
Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic
Francis Borceux, Handbook of Categorical Algebra, Volume 3.
See also:
Last revised on April 19, 2024 at 08:48:34. See the history of this page for a list of all contributions to it.