Sheaves, derived and perverse

Let $X,Y$ be topological spaces and $f:X\to Y$ a continuous map. We let $\Shv(X)$ be the category of sheaves on $X$, $D(\Shv(X))$ the derived category of sheaves on $X$, and $D_b(\Shv(X))$ the bounded variant. Recall that $D(\mathcal A)$ for an abelian category $\mathcal A$ is constructed first by taking $C(\mathcal A)$, the category of cochains of elements of $\mathcal A$, quotienting by chain homotopy, then quotienting by all acylic chains.

Remark: Let $\mathcal F\in \Shv(X)$. Recall:

• a section of $\mathcal F$ is an element of $\mathcal F(U)$ for some $U\subseteq X$,
• a germ of $\mathcal F$ at $x\in X$ is an equivalence class in $\{s\in \mathcal F(U)\ :\ U\owns x\}/\sim_x$,
• $s\sim_x t$ iff every neighborhood $W$ of $x$ in $U\cap V$ has $s|_W = t|_W$, for $s\in \mathcal F(U)$, $t\in \mathcal F(V)$,
• the support of the section $s\in\mathcal F(U)$ is $\supp(s) = \{x\in U\ :\ s \nsim_x 0\}$,
• the support of the sheaf $\mathcal F$ is $\supp(\mathcal F) = \{x\in X\ :\ \mathcal F_x\neq 0\}$.

Definition: The map $f$ induces functors between categories of sheaves, called
\[\begin{array}{r r c l}
\text{direct image} & f_*\ :\ \Shv(X) & \to & \Shv(Y), \\
& (U\mapsto \mathcal F(U)) & \mapsto & (V\mapsto \mathcal F(f^{-1}(V))),\\[15pt]
\text{inverse image} & f^*\ :\ \Shv(Y) & \to & \Shv(X), \\
& (V\mapsto \mathcal G(V)) & \mapsto & \text{sh}\left(U\mapsto \text{colim}_{V\supseteq f(U)} \mathcal G(V)\right),\\[15pt]
\text{direct image with compact support} & f_!\ :\ \Shv(X) & \to & \Shv(Y), \\
& (U\mapsto \mathcal F(U)) & \mapsto & \left(V\mapsto \left\{ s\in\mathcal F(f^{-1}(V))\ :\ f|_{\supp(s)} \text{ is proper}\right\}\right).
\end{array}\]

Above we used that $f:X\to Y$ is proper if $f^{-1}(K)\subseteq X$ is compact, for every $K\subseteq Y$ compact. Next, recall that a functor $\varphi:\mathcal A\to \mathcal B$ induces a functor $R\varphi:D(\mathcal A)\to D(\mathcal B)$, called the (first) derived functor of $\varphi$, given by $R\varphi(A^\bullet) = H^1(\varphi(A)^\bullet)$.

Remark: Each of the maps $f_*,f^*,f_!$ have their derived analogues $Rf_*, Rf^*,Rf_!$, respectively. For reasons unclear, $Rf_!$ has a right adjoint, denoted $Rf^!:D(\Shv(Y))\to D(\Shv(X))$. This is called the exceptional inverse image.

We are now ready to define perverse sheaves.

Definition: Let $A^\bullet \in D(\Shv(X))$. Then:

• the $i$th cohomology sheaf of $A^\bullet$ is $H^i(A^\bullet) = \ker(d^i)/\im(d^i)$,
• $A^\bullet$ is a constructible complex if $H^i(A^\bullet)$ is a constructible sheaf for all $i$,
• $A^\bullet$ is a perverse sheaf if $A^\bullet\in D_b(\Shv(X))$ is constructible and $\dim(\supp(H^{-i}(P))) \leqslant i$ for all $i\in \Z$ and for $P=A^\bullet$ and $P=(A^\bullet)^\vee = (A^\vee)^\bullet$ the dual complex of sheaves.

We finish off with an example.

Example: Let $X = \R$ be a stratified space, with $X_0=0$ the origin and $X_1 = \R\setminus 0$. Let $\mathcal F\in \Shv(X)$ be an $\R$-valued sheaf given by $\mathcal F(U) = \inf_{x\in U} |x|$, and define a chain complex $A^\bullet$ in the following way:
\[0 \longrightarrow A^{-1} = \mathcal F \xrightarrow{ d^{-1}=\text{id} } A^0 = \mathcal F \xrightarrow{ d^0=0 } 0.\]
Note that for any $U\subseteq \R$, we have $H^{-1}(A^\bullet)(U) = \ker(d^{-1})(U) = \ker(\id:\mathcal F(U)\to \mathcal F(U)) = \emptyset$ if $0\not\in U$, and $0$ otherwise. Hence $\supp(H^{-1}(A^\bullet)) = \R\setminus 0$, whose dimension is 1. Next, $H^0(A^\bullet)(U) = \ker(d^0)(U)/\im(d^{-1})(U) = \ker(0:\mathcal F(U)\to 0)/\im(\id:\mathcal F(U)\to \mathcal F(U)) = \mathcal F(U)/\mathcal F(U) = 0$, and so $\dim(\supp(H^0(A^\bullet))) = 0$. Note that $A^\bullet$ is self-dual and constructible, as the cohomology sheaves are locally constant. Hence $A^\bullet$ is a perverse sheaf.

References: Bredon (Sheaf theory, Chapter II.1), de Catalado and Migliorini (What is... a perverse sheaf?), Stacks project (Articles "Supports of modules and sections" and "Complexes with constructible cohomology")

Constructible sheaves

Let $X$ be a topological space with an open cover $\mathcal U = \{U_i\}$, and category $Op(X)$ of open sets of $X$. The goal is to define constructible sheaves and consider some applications. Thanks to Joe Berner for helpful pointers in this area.

Definition: Constructible subsets of $X$ are the smallest family $F$ of subsets of $X$ such that

• $Op(X)\subset F$,
• $F$ is closed under finite intersections, and
• $F$ is closed under complements.

This idea can be applied to sheaves. Recall that a locally closed subset of $X$ is the intersection of an open set and a closed set.

Definition: A sheaf $\mathcal F$ over $X$ is constructible if there exists, equivalently,

• a filtration $\emptyset=U_0\subset \cdots \subset U_n=X$ of $X$ by opens such that $\mathcal F|_{U_{i+1}\setminus U_i}$ is constant for all $i$, or
• a cover $\{V_i\}$ of locally closed subsets of $X$ such that $\mathcal F|_{V_i}$ is constant for all $i$.

Since the category of abelian sheaves over a topological space has enough injectives, we may consider an injective resolution of a sheaf $\mathcal F$ rather than the sheaf itself. The resolution may be considered as living inside the derived category of sheaves on $X$.

Definition: Let $A$ be an abelian category.

• $C(A)$ is the category of cochain complexes of $A$, 
• $K(A) = C(A)$ modulo cochain homotopy, and
• $D(A) = K(A)$ modulo $F\in K(A)$ such that $H^n(F)=0$ for all $n$, called the derived category of $A$.

Next we consider an example. Recall the Ran space $\Ran(M) = \{X\subset M\ :\ 0<|X|<\infty\}$ of non-empty finite subsets of a manifold $M$ and the Čech complex of radius $t>0$ of $P\in \Ran(M)$, a simplicial complex with $n$-cells for every $P'\subset P$ of size $n+1$ such that $d(P'_1,P'_2)<t$ for all $P'_1,P'_2\in P'$.

Example: Consider the subset $\Ran^{\leqslant 2}(M) = \{X\subset M\ :\ 1\leqslant |X|\leqslant 2\}$ of the Ran space. Decompose $X=\Ran^{\leqslant 2}(M)\times \R_+$ into disjoint sets $U_\alpha\cup U_\beta$, where
\[
U_\alpha = \underbrace{\left(\Ran^1(M)\times \R_+\right)}_{U_{\alpha,1}} \cup \underbrace{\bigcup_{P\in \Ran^2(M)}\{P\}\times (d_M(P_1,P_2),\infty)}_{U_{\alpha,2}},
\hspace{1cm}
U_\beta = \bigcup_{P\in \Ran^2(M)} \{P\} \times (0,d_M(P_1,P_2)],
\]
with $d_M$ the distance on the manifold $M$. The idea is that for every $(P,t)\in U_\alpha$, the Čech complex of radius $t$ on $P$ has the homotopy type of a point, whereas on $U_\beta$ has the homotopy type of two points. With this in mind, define a constructible sheaf $F\in\text{Shv}(\Ran^{\leqslant 2}(M)\times \R_+)$ valued in simplicial complexes, with $F|_{U_\alpha}$ and $F|_{U_\beta}$ constant sheaves. Set
\[
F_{(P,t)\in U_\alpha} = F(U_\alpha) = \left(0\to \{*\} \to 0\right),
\hspace{1cm}
F_{(P,t)\in U_\beta} = F(U_\beta) = \left(0\to \{*,*\}\to 0\right).
\]
Note that the chain complex $F(U_\alpha)$ is chain homotopic to $0\to \{-\}\to \{*,*\}\to 0$, where $-$ is a single 1-cell with endpoints $*,*$. To show that this is a constructible sheaf, we need to filter $\Ran^{\leqslant 2}(M)\times \R_+$ into an increasing sequence of opens. For this we use a distance on $\Ran^{\leqslant 2}(M)\times \R_+$, given by $d((P,t),(P',t'))=d_{\Ran(M)}(P,P')+d_\R(t,t'),$ where $d_\R(t,t')=|t-t'|$ and
\[
d_{\Ran(M)}(P,P')=\max_{p\in P}\left\{\min_{p'\in P'}\left\{d_M(p,p')\right\}\right\} + \max_{p'\in P'}\left\{\min_{p\in P}\left\{d_M(p,p')\right\}\right\}.
\]
Note that $U_\alpha$ is open. Indeed, for $(P,t)\in U_{\alpha,1}$, every other $P'\in \Ran^1(M)$ close to $P$ is also in $U_{\alpha,1}$, and if $P'\in \Ran^2(M)$ is close to $P$, then the non-zero component $t\in\R_+$ still guarantees the same homotopy type. The set $U_{\alpha,2}$ is open as well, so $U_\alpha$ is open. The whole space is open, so a filtration $\emptyset\subset U_\alpha\subset X$ works for us.

References: Hartshorne (Algebraic geometry, Section II.3), Hartshorne (Residues and Duality, Chapter IV.1), Kashiwara and Schapira (Sheaves on manifolds, Chapters 2 and 8), Lurie (Higher algebra, Section 5.5.1)