Sheaves
Table of Contents
1. Coverage
A coverage is the minimal notion required to define sheaves. For every object the coverage tells use how we can cover the object (using a family of open embeddings) and the covering families must be stable under pullbacks.
Let \( \mathcal C \) be a category. A coverage on \( \mathcal C \) consists of
- a collection of families \( \{ f_i \colon X_i \rightarrow X \} _{i \in I}\) with target \( X \) for every object \( X \in \mathcal C \) (a covering family or a cover of \( X \)),
such that
If \( \{f_i \} \) is a covering family of \( X \) and \( g \colon Y \rightarrow X \) is a morphism, then there exists a covering family \( h_j \) of \( Y \) such that each composite \( gh_j \) factors through some \( f_i \). To be more precise: For every covering family \( f_i \) of \( X \) and morphism \( g \colon Y \rightarrow X \), there exists a covering family \( h_j \) of \( Y \) such that for every \( j \in J \) there exists \( i \in I \) and a morphism \( k \colon Y_j \rightarrow X_i \) such that
\begin{equation*} \xymatrix{ Y_j \ar[r]^{k} \ar[d]_{h_j} & X_i \ar[d]^{f_i} \\ Y \ar[r]^g & X } \end{equation*}commutes.
We might think of a morphism in a covering family as an open embedding.
Let \( \mathcal C \) be a category. A normal coverage on \( \mathcal C \) is a coverage on \( \mathcal C \) such that for every object \( X \) the identity \( 1_X \) is a cover of \( X \).
Note that every coverage can be turned into a normal coverage by adding all identities to the collection of covering families without changing the category of sheaves wrt that coverage. Therefore in most cases we can assume that our coverage is normal.
Let \( \mathcal C \) be a category. A cartesian coverage on \( \mathcal C \) is a coverage on \( \mathcal C \) such that
- pullbacks of covering families exists along any morphism in \( \mathcal C \),
- the pullback of a covering family is a covering family.
1.0.1. TODO colimit induced by a cover in a cartesian coverage
Colimit induced by a cover in a cartesian coverage.
Let \( \mathcal C \) be a category. A subcanonical coverage on \( \mathcal C \) is a coverage on \( \mathcal C \) such that
- every representable functor \( \mathcal C(-,c) \colon \mathcal C ^{op} \rightarrow \Set \) is a sheaf.
Let \( \mathcal C \) be a category. A standard coverage on \( \mathcal C \) is a normal, cartesian, and subcanonical coverage on \( \mathcal C \).
2. Site
A site is a category equipped with a coverage.
A normal site is a category equipped with a normal coverage. A cartesian site is a category equipped with a cartesian coverage. A subcanonical site is a category equipped with a subcanonical coverage. A standard site is a category equipped with a standard coverage.
3. Sheaf
The notion of a sheaf is very general, like the notion of a topos it has several interpretations, or how we call it, there are multiple intuitive models of the notion “sheaf”. First we give an elementary definition using sets (this definition needs the least assumptions on the source category but the target category has to be \( \Set \) or maybe a topos). Next we give a categorical definition (this definition needs more assumptions on the source category but the target category can be any category). Then we discuss intuitive models for sheaves.
3.1. Elementary (set theoretic) Definition of Sheaves
Let \( \mathcal C \) be a site. A sheaf on \( \mathcal C \) is
- a functor \( F \colon \mathcal C ^{op} \rightarrow \Set \) (a presheaf),
such that the sheaf condition holds:
- For every covering family \( \{f_i \colon U_i \rightarrow U \} \) in \( \mathcal C \), for every family \( (x_i)_{i \in I} \in \prod _{i \in I} X(U_i) \) with for all \( g \colon V \rightarrow U_i \) and \( h \colon V \rightarrow U_j \) with \( f_i g = f_j h \) holds \( X(g)(x_i) = X(h)(x_j) \in X(V) \), there exists an unique \( x \in X(U) \) with \( X(f_i)(x) = x_i \) for every \( i \in I \).
Let \( \mathcal C \) be a category. Let \( X \in \PSh(\mathcal C)\) be a presheaf and \( (f_i \colon U_i \rightarrow U) _{i \in I}\) a family in \( \mathcal C \). The corresponding restriction map is \( \Res \colon X(U) \rightarrow \prod _{i \in I}X(U_i) \) given by
\begin{equation*} x \mapsto X(f_i)(x) = x \circ f_i. \end{equation*}Now let \( \mathcal C \) be a site, \( X \in \Sh(\mathcal C) \) a sheaf and \( (f_i \colon U_i \rightarrow U) _{i \in I}\) a cover in \( \mathcal C \). Then the sheaf condition says precisely that the restriction map \( \Res \colon X(U) \rightarrow \prod _{i \in I}X(U_i) \) has an inverse. We call this inverse the corresponding gluing map \( \Glue \colon \prod _{i \in I }X(U_i) \rightarrow X(U) \).
In the words the sheaf condition says: every compatible family \( (x_i)_{i \in I} \in \prod _{i \in I} X(U_i) \) corresponds to a unique element in \( X(U)\).
3.2. Abstract Definition of Sheaves
The sheaf condition in set theoretic terms illustrates where the intuition of “maps into an object that can be glued” but we want to give a categorical definition that can make sense in other categories than \( \Set \) and still gives the set theoretic definition for \( \Set \).
3.3. Sheaves as Generalized Spaces
Let \( \mathcal C \) be a site. Then we can think of a sheaf on \( \mathcal C \) as a generalized space that is probable by the local models in \( \mathcal C \) where the probes are compatible with the coverings in \( \mathcal C \). Remember that we can think a presheaf as a generalized object, that is probable by the objects in \( \mathcal C \). Indeed let \( X \colon \mathcal C ^{op} \rightarrow \Set \) be a presheaf. By the Yoneda lemma, \( X \) precisely consists of the probing data for every object in \( \mathcal C \). The categorical manifestation of this is that \( X \) is the colimit over all morphisms from objects in \( \mathcal C \) to \( X \) (over all probes). Hence if we think of objects in \( \mathcal C \) as local models of spaces then we think of a presheaf as a naive generalized space. Indeed a presheaf is a sheaf with respect to the empty coverage. But if our category of local models is equipped with more information, namely how we can cover a local model using other local models (a coverage), then a probe from an object \( U \in \mathcal C \) to a generalized space \( X \) should already be determined by (compatible) probes from a covering of \( U \) to \( X \). Hence we arrive a the sheaf condition.