The Goal / An Informal Introduction
Loosely speaking, the definition for sheaf many of us first learn is that it is a presheaf, , which satisfies some glueing condition: for each open subset of our fixed topological space , and open covering , sections of are precisely tuples of sections which agree on their restrictions to .
Formally, the above condition for a presheaf to be a sheaf is often described by saying that is the equalizer (or limit) of the following diagram:
Going one step further, and rewriting this limit using some category theory facts, which will be more carefully described in this post, we see that being a sheaf can be expressed as an isomorphism of hom sets in presheaves:
Once one learns about sheaves, a possible eventual question is “what is the process for obtaining sheaves?” for which the answer is usually called sheafification. It is well documented that if you take the colimit of the right hand side of the previous isomorphism over all possible open covers, you obtain what is often referred to as . This presheaf is not necessarily a sheaf but we can finally obtain a sheaf by applying this construction a second time.
Alternatively, we could consider a similar colimit but over all hypercovers and obtain a sheaf in one iteration. Unpacking the previous sentence is the content of this entire note. Personally, the goal for me to rewrite a known story in this language is so that I can apply similar ideas in order to sheafify simplicial presheaves. This story is also well known but the explicit details seem to be missing from the literature.
Consider two simplicial complexes and as pictured below. asdf
Definition: A sieve in a category is a full subcategory of closed under precomposition by morphisms in .https://ncatlab.org/nlab/show/sieve
In other words, if is a seive with a morphism in , and is any morphism in then must also be a morphism in .
Definition: A sieve on an object in a category is a sieve in the over category .https://ncatlab.org/nlab/show/sieve
So now a sieve on an object has as its objects, morphisms to , and as its morphisms, commutative triangles closed under precomposition:
Next we look at the definitions of Grothendieck (pre)topology where the prototypical example I have in mind is that which is induced by open subsets of a fixed topological space, . As I will generally have in mind the pretopology, we can start with that definition first.
Definition: A Grothendieck pretopology on a category is an assignment to each object in , a collection of distinguished families of maps which we call covering families, satisfying the following
(PT0) (Fibered Products) If is in with a member of some covering family, and is any map in then the fibered product exists in .
(PT1) (Stable Base Change) For any object in , any morphism in , and any covering family , the family of maps is a covering family.
(PT2) (Local Character) If is a covering family and for each , we have a covering family , then the family of all compositions is a covering family.
(T3) (Identity) Whenever we have an isomorphism in then the singleton collection containing just this isomorphism is a covering family of .https://en.wikipedia.org/wiki/Grothendieck_topology
So for the example where our category is the collection of all subsets of a topological space where the morphisms are given by inclusion, we can consider objects as subspaces of and our covering families are open coverings of the subspace coming from the subspace topology. (PT0) Says that we can intersect two subspaces of and obtain another subspace of . (PT1) Says that if we have an open covering of a subspace , and we consider a subspace of , then the collection of intersections is an open covering of . (PT2) Says that if we start with an open covering and cover each component , then the entire collection ranging over all components is another open covering of . Finally (PT3) says that the singleton containing is an open cover of itself.
In many classical and more generalized settings, you might see the use instead of a Grothendieck topology, without referring to these families and instead focused on sieves. We state the definition and then again look at how it applies to our toy example.
Definition: A Grothendieck topology on a category consists of, for each object in , a collection of distinguished sieves labeled which we call covering sieves, satisfying the following
(T1) (Base Change) If is a covering sieve on and is a map in then is a covering sieve on .
(T2) (Local Character) Let be a covering sieve on and be any sieve on , so that for each and , is a covering sieve on , then is a covering sieve on .
(T3) (Identity) The collection of all maps in with codomain is a covering sieve on .
A category with a chosen Grothendieck topology is often referred to as a site.https://en.wikipedia.org/wiki/Grothendieck_topology
Using again the example where we use the category of subsets of a topological space , we can see that the covering families of a subspace do not form sieves, and thus can not be covering sieves. To see this just take an open cover , and then consider a non-open subset and note that the map was not in our original covering family, so it was not closed under precomposition and thus can not be a sieve.
Instead, we would like to take a pretopology, and from it generate a topology on a category.
Pretopology ⤳ Topology
Construction: Given a pretopology on a category define the associated topology by distinguishing any sieve which contains a covering family to be updgraded to a covering sieve.
So now given a pretopology on described by covering families on objects, a sieve will be considered a covering sieve if every map factors through some coming from