A few questions based on the definitions at the end:
- The meaning of $a \rightsquigarrow b$ (#7 in abstract simplical complex definition). Not sure what "that is compatible with the inclusions of faces" means.
- If the sheaf vector spaces can be different types among the faces. Or they all have to be the same type (#1 in first sheaf definition).
- If an abstract simplical complex is an open cover or related. The second sheaf definition says a sheaf is on an open cover, while the first says it's on an abstract simplical complex.
- What the inverse limit part is saying (the second sheaf definition at the bottom): $$F(Q)\to\varprojlim_{I\in N(Q)}F(U_I) =: F[Q]$$
The definitions of sheaf are from here.
An abstract simplical complex $X$ is:
- Of a set $A$
- A collection $X$ of subsets of $A$
- Every element $x$ of $X$ (subset of $A$) implies that every subset of $x$ is in $X$.
- Every $x$ in $X$ with $k+1$ elements is a k-face of $X$. This is the dimension.
- 0-dimensional faces are vertices.
- 1-dimensional faces are edges.
- A face $a$ "includes onto" $b$ ($a \rightsquigarrow b$) whenever $a$ is a proper subset of $b$.
A sheaf $F$ on an abstract simplical complex $X$ consists of:
- A vector space $F(a)$ to each face of a of $X$ (the stalk at $a$).
- A linear map $F(a \rightsquigarrow b) : F(a) \to F(b)$ (the restriction along $a \rightsquigarrow b$).
- $F(b \rightsquigarrow c) \circ F(a \rightsquigarrow b) = F(a \rightsquigarrow c)$ whenever $a \rightsquigarrow b \rightsquigarrow c$.
Given a simplicial complex, a sheaf is merely the assignment of vector-valued data to each face that is compatible with the inclusions of faces.
So to each face we assign a vector space. The face is a vertex of a graph if the face is 0-dimensional. The face is an edge of the graph if the face is 1-dimensional. So we assign vector space to each node and/or edge (and/or more if there's higher dimension). I think I understand that part. I don't understand the "is compatible with the inclusion of faces" part.
From the full book, the definitions are:
A sheaf is a functor that commutes with limits coming from open covers.
Suppose $Λ = \{1, \dotsc , n\}$ is a finite discrete category.
The open set category is:
- Associated to a topological space $X$
- Denoted $\mathbf{Open}(X)$
- Whose objects are the open sets of $X$
- And whose morphisms $U \to V$ are for each pair related by inclusion $U \subseteq V$.
An open cover $Q$ is:
- Given a topological space $X$
- And an open set $U$ on $X$
- It's a collection of open sets on the open set $U$, $Q = \{U_i\}_{i\in Λ}$ whose union is $U$.
The nerve $N(Q)$ of an open cover $Q$:
- Gives us an abstract simplical complex $N(Q)$
- Elements are subsets of $Q$, $I = \{i_0, \dots ,i_n\}$
- (for which $U_I := U_{i_0} \cap \dots \cap U_{i_n} \neq \emptyset$
A pre-sheaf is:
- A functor $F : \mathbf{Open}(X)^{op} \to \mathbf{D}$
A pre-cosheaf is:
- A functor $\hat{F} : \mathbf{Open}(X) \to \mathbf{D}$
A sheaf $F$ is
- On $Q$ (open cover).
- If the unique map from $F(U)$ to the limit of $F \circ \rho^{op}_Q$ is an isomorphism, written: $$F(Q)\to\varprojlim_{I\in N(Q)}F(U_I) =: F[Q]$$