The Goal / An Informal Introduction

Loosely speaking, the definition for sheaf many of us first learn is that it is a presheaf, F, in which satisfies some glueing condition: for each open subset U of our fixed topological space X, and open covering \{U_i \}, sections of F(U) are precisely tuples of sections s_i \in F(U_i) which agree on their restrictions to U_i \cap U_j.

Formally, the above condition for a presheaf to be a sheaf is often described by saying that F(U) 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 F^{\dagger}(X). 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

Sieves & Grothendieck Topologies

Definition: A sieve in a category \mathcal{C} is a full subcategory of \mathcal{C} closed under precomposition by morphisms in \mathcal{C}.

https://ncatlab.org/nlab/show/sieve

In other words, if \mathcal{A} \hookrightarrow \mathcal{C} is a seive with V \to X a morphism in \mathcal{A}, and U \to V is any morphism in \mathcal{C} then U \to V \to X must also be a morphism in \mathcal{A}.

Definition: A sieve on an object X in a category \mathcal{C} is a sieve in the over category \mathcal{C}/X.

https://ncatlab.org/nlab/show/sieve

So now a sieve on an object X has as its objects, morphisms to V_i \to X, 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, X. As I will generally have in mind the pretopology, we can start with that definition first.

Definition: A Grothendieck pretopology on a category \mathcal{C} is an assignment to each object U in \mathcal{C}, a collection of distinguished families of maps \{ U_i \to U\} which we call covering families, satisfying the following

(PT0) (Fibered Products) If U is in \mathcal{C} with U_i \to U a member of some covering family, and V \to U is any map in \mathcal{C} then the fibered product U_i \times_X V exists in $\mathcal{C}$.

(PT1) (Stable Base Change) For any object U in \mathcal{C}, any morphismV \to U in \mathcal{C}, and any covering family \{ U_i \to V\}, the family of maps \{U_i \times_X V \to V\} is a covering family.

(PT2) (Local Character) If \{U_i \to U\} is a covering family and for each i, we have a covering family \{ U_{i,j} \to U_i\}, then the family of all compositions \{ U_{i,j} \to U_i \to U\} is a covering family.

(T3) (Identity) Whenever we have an isomorphism V \xrightarrow{\sim} U in \mathcal{C} then the singleton collection containing just this isomorphism is a covering family of U.

https://en.wikipedia.org/wiki/Grothendieck_topology

So for the example where our category \mathcal{C} = \mathcal{P}(X) is the collection of all subsets of a topological space X where the morphisms are given by inclusion, we can consider objects U as subspaces of X and our covering families \{U_i \to U\} are open coverings of the subspace U coming from the subspace topology. (PT0) Thus means that we can intersect two subspaces of X and obtain another subspace of X. (PT1) Means that if we have an open covering \{U_i \to U\} of a subspace U, and we consider a subspace V of U, then the collection of intersections \{U_i \cap V = U_i \times_X V \to V is an open covering of V. (PT2) Says that if we start with an open covering \{U_i \to U\} and cover each component U_i, then the entire collection ranging over all components is another open covering of U. Finally (PT3) says that the singleton containing U 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 \mathcal{J} on a category \mathcal{C} consists of, for each object U in \mathcal{C}, a collection of distinguished sieves labeled \mathcal{J}(U) which we call covering sieves, satisfying the following

(T1) (Base Change) If S is a covering sieve on U and V \xrightarrow{f} U is a map in \mathcal{C} then f^* S:= \{ W \xrightarrow{g} V \ \vert \ (W \xrightarrow{g} U \xrightarrow{f} U) \in S \} is a covering sieve on V.

(T2) (Local Character) Let S be a covering sieve on U and T be any sieve on U, so that for each V \in \mathcal{C} and (V \xrightarrow{f} U) \in S, f^*T is a covering sieve on V, then T is a covering sieve on U.

(T3) (Identity) The collection of all maps in \mathcal{C} with codomain U is a covering sieve on U.

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 \mathcal{P}(X) of subsets of a topological space X, we can see that the covering families \{U_i \to U\} of a subspace do not form sieves, and thus can not be covering sieves. To see this just take an open cover \{U_i \to U\}, and then consider a non-open subset A \to U_i and note that the map A \to U_i \to U 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 \mathcal{C} 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 \mathcal{C} described by covering families on objects, a sieve mathcal{S} will be considered a covering sieve if every map (V \to U) \in \mathcal{S} factors through some U_i \to U coming from

Under construction…Come back soon

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