Artin gluing a sheaf 2: simplicial sets and configuration spaces

The goal of this post is to extend the previous stratifying map to simplicial sets, and to generalize the sheaf construction to $X = \Conf_n(M)\times \R_{\geqslant 0}$ for arbitrary integers $n$, where $M$ is a smooth, compact, connected manifold. We work with $\Conf_n(M)$ instead of $\Ran^{\leqslant n}(M)$ because Lemma 1 and Proposition 2 have no chance of extending to $\Ran^{\leqslant n}(M)$ without major modifications (see Remark 3 at the end of this post).

Recall $SC$ is the category of simplicial complexes and simplicial maps, with $SC_n$ the full subcategory of simplicial complexes on $n$ vertices. Our main function is \[ \begin{array}{r c c c l}
f\ :\ X & \tov{f_1} & SC & \tov{f_2} & \sSet, \\
(P,a) & \mapsto & VR(P,a) & \mapsto & \Hom_{\Set}(\Delta^\bullet,VR(P,a)).
\end{array} \] On $\Conf_n(M)$ we have a natural metric, the Hausdorff distance $d_H(P,Q) = \max_{p\in P}\min_{q\in Q}d(p,q)+\max_{q\in Q}\min_{p\in P}d(p,q)$. This induces the 1-product metric on $X$, as \[ d_X((P,a),(Q,b)) = d_H(P,Q) + d(a,b), \] where $d$ without a subscript is Euclidean distance. We could have chosen any other $p$-product metric, but $p=1$ makes computations easier. For a given $(P,t)\in X$, write $P = \{P_1,\dots,P_n\}$ and define its maximal neighborhood to be the ball $B_X(\min\{\delta_1,\delta_2,t\},P)$, where \[ \delta_1 = \min_{i<j}\{d(P_i,P_j)\},
\hspace{1cm}
\delta_2 = \min_{i<j}\{|d(P_i,P_j)-t|\ :\ d(P_i,P_j)\neq t\}. \]

Lemma 1: Any path $\gamma:I\to X$ induces a unique morphism $f(\gamma(0))\to f(\gamma(1))$ of simplicial sets.

Proof: Write $\gamma(0) = \{P_1,\dots,P_n\}$ and $\gamma(1) = \{Q_1,\dots,Q_n\}$. The map $\gamma$ induces $n$ paths $\gamma_i:I\to M$ for $i=1,\dots,n$, with $\gamma_i$ the path based at $P_i$. Let $s:\gamma(0)\to \gamma(1)$ be the map on simplicial complexes defined by $P_i\mapsto \gamma_i(1)$. Since we are in the configuration space, where points cannot collide (as opposed to the Ran space), this is a well-defined map. Then $f_2(s)$ is a morphism of simplicial complexes. $\square$

Note the morphism of simplicial sets induced by any path in a maximal neighborhood of $x\in X$ is the identity morphism. We now move to describing a sheaf over all of $X$.

Definition: Let $X$ be any topological space and $\mathcal C$ a category with pullbacks. Let $A\subseteq X$ open and $B=X\setminus A \subseteq X$ closed, with $i:A\hookrightarrow X$ and $j:B\hookrightarrow X$ the inclusion maps. Let $\mathcal F$ be a $\mathcal C$-valued sheaf on $A$ and $\mathcal G$ a $\mathcal C$-valued sheaf on $B$. Then the \emph{Artin gluing} of $\mathcal F$ and $\mathcal G$ is the $\mathcal C$-valued sheaf $\mathcal H$ on $X$ defined as the pullback, or fiber product, of $i_*\mathcal F$ and $j_*\mathcal G$ over $j_*j^*i_*\mathcal F$ in the diagram below.

Note the definition requires a choice of sheaf map $\varphi:\mathcal G\to j^*i_*\mathcal F$. In the proof below, this sheaf map will be the morphism of simplicial sets from Lemma 1 through the functor $\Hom_\Set(\Delta^\bullet,-) = f_2(-)$.

Recall the ordering of $SC_n$ described by the only definition in a previous post ("Exit paths, part 2," 2017-09-28). Fix a cover $\{A_i\}_{i=1}^{N}$ of $SC_n$ by nested open subsets (so $N=|SC_n|$), with $B_i := f_1^{-1}(A_i)$ and $B_{\leqslant i} := \bigcup_{j=1}^i B_i$. We now have an induced order on and cover of $\im(f)=\sSet'$, as a full subcategory of $\sSet$. Even more, we now have an induced total order on $\sSet' = \{S_1,\dots,S_N\}$, with $S_i$ the unique simplicial set in $A_i\setminus A_{i-1}$. For example, $S_1=\Hom_\Set(\Delta^\bullet,\Delta^n)$ and $S_{N}=\Hom_\Set(\Delta^\bullet,\bigcup_{i=1}^n\Delta^0)$.

For ease of notation, we let $B_0 = \emptyset$ and write $S_\emptyset = \Hom(\Delta^\bullet,\emptyset)$, $S_0 = \Hom(\Delta^\bullet,\Delta^0)$.

Definition 1: Let $\mathcal F_i:\Op(B_i)^{op}\to \sSet$ be the locally constant sheaf given by $\mathcal F_i(U_x) = S_i$, where $U_x$ is a subset of the maximal neighborhood of $x\in B_i$. In general, \[ \mathcal F_i(U) = \begin{cases}
S_i & \text{ if }\begin{array}[t]{l}U\neq \emptyset, \\U\text{ is path connected},\\\text{every loop }\gamma:I\to U\text{ induces }\id:f(\gamma(0))\to f(\gamma(1)),\end{array} \\
S_\emptyset & \text{ else if }U\neq\emptyset, \\
S_0 & \text{ else.}
\end{cases} \] In general, we say $U\subseteq X$ is good if it is non-empty, path connected, and every loop $\gamma:I\to U$ induces the identity morphism on simplicial sets.

Proposition 2: Let $\mathcal F_{\leqslant 1} = \mathcal F_1$, and $\mathcal F_{\leqslant i}$ be the sheaf on $B_{\leqslant i}$ obtained by Artin gluing $\mathcal F_i$ onto $\mathcal F_{\leqslant i-1}$, for all $i=2,\dots,N$. Then $\mathcal F = \mathcal F_{\leqslant N}$ is the $SC_n$-constructible sheaf on $X$ described by \[ \mathcal F(U) = \begin{cases}
S_{\max\{1\leqslant \ell\leqslant N\ :\ U\cap B_{\ell}\neq \emptyset\}} & \text{ if $U$ is good,}\\
S_\emptyset & \text{ else if }U\neq\emptyset, \\
S_0 & \text{ else.}
\end{cases} \hspace{2cm} (1) \]

Proof: We proceed by induction. Begin with the constant sheaf $\mathcal F_1$ on $B_1$ and $\mathcal F_2$ on $B_2$, which we would like to glue together to get a sheaf $\mathcal F_{\leqslant2}$ on $B_{\leqslant 2}$. Since $f_1$ is continuous in the Alexandrov topology on the poset $SC_{\leqslant n}$, $B_1\subseteq B_{\leqslant 2}$ is open and $B_2 \subseteq B_{\leqslant 2}$ is closed. Let $i:B_1\hookrightarrow B_{\leqslant 2}$ and $j:B_2\hookrightarrow B_{\leqslant 2}$ be the inclusion maps. The sheaf $j^*i_*\mathcal F_1$ has support $\closure(B_1)\cap B_2 \neq \emptyset$ with \[ j^*i_*\mathcal F_1(U) = \colim_{V\supseteq j(U)}\left[i_*\mathcal F_1(V)\right] = \colim_{V\supseteq U}\left[\mathcal F_1(V\cap B_1)\right] = \begin{cases}
S_1 & \text{ if }U\cap \closure(B_1)\text{ is good}, \\ S_\emptyset & \text{ else},
\end{cases} \] for any non-empty $U\subseteq B_2$. Let the sheaf map $\varphi:\mathcal F_2\to j^*i_*\mathcal F_1$ be the inclusion simplicial set morphism on good sets (it can be thought of as induced through Lemma 1 by a path starting in $U\cap B_2$ and ending in $V\cap B_1$, for $V$ a small enough set in the colimit above). Note that $S_2 = \Hom_\Set(\Delta^\bullet,\Delta^n\setminus \Delta^1)$, where $\Delta^n\setminus \Delta^1$ is the simplicial complex resulting from removing an edge from the complete simplicial complex on $n$ vertices. Let $\mathcal F_{\leqslant 2}$ be the pullback of $i_*\mathcal F_1$ and $j_*\mathcal F_2$ along $j_*j^*i_*\mathcal F_1$, and $U\subseteq B_{\leqslant 2}$ a good set. If $U\subseteq B_1$, then $\mathcal F_{\leqslant 2}(U) = \mathcal F_1(U)=S_1$, and if  $U\subseteq B_2$, then $\mathcal F_{\leqslant 2}(U) = \mathcal F_2(U) = S_2$. Now suppose that $U\cap B_1 \neq \emptyset$ but also $U\cap B_2\neq\emptyset$, which, since $U$ is good, implies that $U\cap \closure(B_1)\cap B_2\neq\emptyset$. Then we have the pullback square

If $U$ is not good, then the simplicial sets are $S_\emptyset$ or $S_0$, with nothing interesting going on. The pullback over a good set $U$ can be computed levelwise as \[ \mathcal F_{\leqslant 2}(U)_m = \{(\alpha,\beta)\in (S_1)_m\times (S_2)_m\ :\ \alpha=j_*\varphi(\beta)\}. \hspace{2cm} (2)\] Since $j_*\varphi$ is induced by the inclusion $\varphi$, it is the identity on its image. So $\alpha = j_*\varphi(\beta)$ means $\alpha=\beta$, or in other words, $\mathcal F_{\leqslant 2}(U)=S_2$. Hence for arbitrary $U\subseteq B_{\leqslant 2}$, we have \[ \mathcal F_{\leqslant 2}(U) = \begin{cases}
S_{\max\{\ell=1,2\ :\ U\cap B_{\ell}\neq \emptyset\}} & \text{ if $U$ is good,}\\
S_\emptyset & \text{ else if }U\neq\emptyset, \\
S_0 & \text{ else.}
\end{cases}\]

For the inductive step with $k>1$, let $\mathcal F_{\leqslant k}$ be the sheaf on $B_{\leqslant k}$ defined as in Equation (1), but with $k$ instead of $N$. We would like to glue $\mathcal F_{\leqslant k}$ to $\mathcal F_{k+1}$ on $B_{k+1}$ to get a sheaf $\mathcal F_{\leqslant k+1}$ on $B_{\leqslant k+1}$. As before, $B_k \subseteq B_{\leqslant k+1}$ is open and $B_{k+1}\subseteq B_{\leqslant k+1}$ is closed. For $i:B_k\hookrightarrow B_{\leqslant k+1}$ and $j:B_{k+1}\hookrightarrow B_{\leqslant k+1}$ the inclusion maps, the sheaf $j^*i_*\mathcal F_{\leqslant k}$ has support $\closure(B_{\leqslant k})\cap B_{k+1}$, with \[ j^*i_*\mathcal F_{\leqslant k}(U) = \colim_{V\supseteq j(U)}\left[i_*\mathcal F_{\leqslant k}(V)\right] = \colim_{V\supseteq U}\left[\mathcal F_{\leqslant k}(V\cap B_{\leqslant k})\right] = \begin{cases} S_{\max\{1\leqslant \ell\leqslant k\ :\ U\cap \closure(B_\ell)\neq\emptyset\}} & \text{ if }U\cap \closure(B_{\leqslant k})\text{ is good,} \\ S_\emptyset & \text{ else,} \end{cases} \] for any non-empty $U\subseteq B_{k+1}$. Let the sheaf map $\varphi:\mathcal F_{k+1}\to j^*i_*\mathcal F_{\leqslant k}$ be the inclusion simplicial set morphism on good sets (it can be thought of as induced through Lemma 1 by a path starting in $U\cap B_{k+1}$ and ending in $V\cap B_{\leqslant k}$, for $V$ a small enough set in the colimit above). For $U\subseteq B_{\leqslant k+1}$ a good set, if $U\subseteq B_{\leqslant k}$, then $\mathcal F_{\leqslant k+1}(U) = \mathcal F_{\leqslant k}(U)$, and if  $U\subseteq B_{k+1}$, then $\mathcal F_{\leqslant k+1}(U) = \mathcal F_{k+1}(U) = S_{k+1}$. Now suppose that $U\cap B_{\leqslant k} \neq \emptyset$ but also $U\cap B_{k+1}\neq\emptyset$, which, since $U$ is good, implies that $U\cap \closure(B_{\leqslant k})\cap B_{k+1}\neq\emptyset$. Then we have the pullback square

If $U$ is not good, then the simplicial sets are $S_\emptyset$ or $S_0$, with nothing interesting going on. Again, as in Equation (2), the pullback $\mathcal F_{\leqslant k+1}$ on a good set $U$ is \[ \mathcal F_{\leqslant k+1}(U)_m = \{(\alpha,\beta)\in (S_\ell)_m\times (S_{k+1})_m\ :\ \alpha = j_*\varphi(\beta)\}, \] and as before, this implies that $\mathcal F_{\leqslant k+1}(U) = S_{k+1}$. Hence $\mathcal F_{\leqslant k+1}$ is exactly of the form as in Equation (1), with $k+1$ instead of $N$, and by induction we get the desired description for $\mathcal F_{\leqslant N}= \mathcal F$.  $\square$

Remark 3: The statements given in this post do not extend to $\Ran^{\leqslant n}(M)$, at least not as stated. Lemma 1 fails if  somewhere along the path $\gamma$ a point splits in two or more points, as there is no canonical choice which of the "new" points should be the image of the "old" point. This means that the proof of Proposition 2 will also fail, because we relied on a uniquely defined sheaf map $\varphi$ between strata.

Next, we hope to use this approach to describe classic persistent homology results, and maybe link this to the concept of persistence modules.

References: Milne (Etale cohomology, Chapter 2.3)

Artin gluing a sheaf 1: a small example

The goal of this post is to describe a sheaf on a particular stratified space using locally constant sheaves defined on the strata. Thanks to Joe Berner for helpful discussions.

Recall the direct image and inverse image sheaves from a previous post ("Sheaves, derived and perverse," 2017-12-05). Let $M$ be a smooth, compact, connected manifold, and $X = \Ran^{\leqslant 2}(M)\times \R_{\geqslant 0}$. Let $SC$ be the category of abstract simplicial complexes and simplicial maps. All sheaves will be functors $\text{Op}(-)^{op}\to SC$. The space $X$ looks like the diagram below.

Let $Y = A\cup B$. Note that $A\subseteq Y$ is open, $B\subseteq Y$ is closed, $Y\subseteq X$ is open, and $C\subseteq X$ is closed. There is a natural stratified map $f:X\to \{1,2,3\}$, with $\{1,2,3\}$ given the natural ordering. The map $f$ is described by $f^{-1}(3) = A$, $f^{-1}(2) = B$, and $f^{-1}(1) = C$. Define the inclusion maps \begin{align*}
i\ &:\ A \hookrightarrow Y, & k\ &:\ Y\hookrightarrow X,\\
j\ &:\ B \hookrightarrow Y, & \ell\ &:\ C\hookrightarrow X.
\end{align*} Define the following constant sheaves on $A,B,C$, respectively:

If $U = \emptyset$, all three give back the simplicial complex on a single vertex. We will now attempt to define a sheaf on all of $X$ by gluing sheaves on the strata. Choose some subsets of $X$ as below on which to test the sheaves.

Step 1: Extend $\mathcal F$ and $\mathcal G$ to a sheaf on $Y$.

The direct image of $\mathcal F$ via $i$, as a sheaf on $Y$, is

for any $U\subseteq Y$. The inverse image of $i_*\mathcal F$ via $j$, as a sheaf on $B$, is

for any $U\subseteq B$. Note $j^*i_*\mathcal F(B')$ is the 0-simplex and $j^*i_*\mathcal F(B'')$ is the 1-simplex. The inverse image sheaf is actually defined as the sheafification of the presheaf obtained by taking the colimit, but the sheaf axioms are easily seen to be satisfied here, as the support is on a closed subset.

Following the MathOverflow question, we need to define a map $\mathcal G \to j^*i_*\mathcal F$ of sheaves on $B$. Since the support of $j^*i_*\mathcal F$ is only $\text{cl}(A)\cap B$, it suffices to define the map here, and we can do it on stalks. There is a natural simplicial map

which we use as the sheaf map. It seems we should now have a sheaf on all of $Y$ now, but the result is not immediate. Following the proof of Theorem 3.10 in Chapter 2 of Milne, we need to take the fiber product, or pullback, of $i_*\mathcal F$ and $j_*\mathcal G$ over $j_*j^*i_*\mathcal F$, call it $\mathcal K$. Consider the pullback diagram on sets like $B'''$:

Hence it makes sense that $\mathcal K(B''')$ is two 0-simplicies. We now have a sheaf $\mathcal K$ on $Y$ given by

Step 2: Extend $\mathcal K$ and $\mathcal H$ to a sheaf on $X$.

The direct image of $\mathcal K$ via $k$, as  a sheaf on $X$, is

for any $U\subseteq X$. The inverse image of $k_*\mathcal K$ via $\ell$, as a sheaf on $C$, is

for any $U\subseteq C$. We need to again define a map $\mathcal H\to \ell^*k_*\mathcal K$ of sheaves on $C$. On stalks we naturally have maps

due to the fact that both complexes are symmetric, so sending to one or the other vertex is the same. Let $\mathcal L$ be the sheaf we should now have defined over all of $X$, by taking the fiber product of $\ell_*\mathcal H$ and $k_*\mathcal K$ over $\ell_*\ell^*k_*\mathcal K$. Let us consider its pullback diagrams for the sets $L',M',N'$.

It seems that we should set $\mathcal L(L') = \mathcal L(M') = \mathcal L(N')$ to be the 0-simplex. We now have a sheaf $\mathcal L$ on $X$ given by

The next goal is to extend this approach to $\Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$. An immediate difficulty seems to be finding canonical simplicial maps like $\varphi$ and $\psi$, but hopefully a choice of increasing nested open cover of the startifying set of $X$ will solve this problem.

References: MathOverflow (Question 54037), Milne (Etale cohomology, Chapter 2.3)

A naive constructible sheaf

In this post we describe a constructible sheaf over $X=\Ran^{\leqslant n}(M)\times \R_{>0}$ valued in simplicial complexes, for a compact, smooth, connected manifold $M$. We note however that it does not capture all the information about the underlying space. Thanks to Joe Berner for helpful ideas.

Recall the category $SC$ of simplicial complexes and simplicial maps, as well as the full subcategories $SC_n$ of simplicial complexes with $n$ vertices (the vertices are unordered). Let $A = \bigcup_{k=1}^n SC_n$ with the ordering $\leqslant_A$ as in a previous post ("Ordering simplicial complexes with unlabeled vertices," 2017-12-03), and $f:X\to A$ the stratifying map. Let $\{A_k\}_{k=1}^N$ be a cover of $X$ by nested open sets of the type $f^{-1}(U_S) = f^{-1}(\{T\in A\ :\ S\leqslant_A T\})$, whose existence is guaranteed as $A$ is finite. Note that $f(A_1)$ is a singleton containg the complete simplex on $n$ vertices.

Remark: For every simplicial complex $S\in A$, there is a locally constant sheaf over $f^{-1}(S)\subseteq X$. Given the cover $\{A_k\}$ of $X$, denote this sheaf by $\mathcal F_k \in \Shv(A_k\setminus A_{k-1})$ and its value by $S_k\in SC$.

Let $i^1:A_1\hookrightarrow A_2$ and $j^2:A_2\setminus A_1 \hookrightarrow A_2$ be the natural inclusion maps . Note that $A_1$ is open and $A_2\setminus A_1$ is closed in $A_2$. The maps $i^1,j^2$ induce direct image functors on the sheaf categories\[i^1_*:\Shv(A_1) \to \Shv(A_2),
\hspace{1cm}
j^2_*:\Shv(A_2\setminus A_1) \to \Shv(A_2).\]The induced sheaves in $\Shv(A_2)$ are extended by 0 on the complement of the domain from where they come. Note that since $A_2\setminus A_1\subseteq A_2$ is closed, $j^2_*$ is the same as $j^2_!$, the direct image with compact support. We then have the direct sum sheaf $i^1_*\mathcal F_1 \oplus j_*^2\mathcal F_2 \in \Shv(A_2)$, which we interpret as the disjoint union in $SC$. Then\[\left(i_*^1\mathcal F_1 \oplus j_2^*\mathcal F_2\right)(U) = \begin{cases}
S_1 & \text{ if }U\subseteq A_1, \\
S_2 & \text{ if }U\subseteq A_2\setminus A_1, \\
S_1\sqcup S_2 & \text{ else,}
\end{cases}
\hspace{1cm}
\left(i_*^1\mathcal F_1 \oplus j_2^*\mathcal F_2\right)_{(P,t)} = \begin{cases}
S_1 & \text{ if } (P,t)\in A_1, \\
S_2 & \text{ if }(P,t)\in \text{int}(A_2\setminus A_1), \\
S_1\sqcup S_2 & \text{ else,}
\end{cases}\]for $U\subseteq A_2$ open and $(P,t)\in A_2$. Generalizing this process, we get a sheaf on $X$. The diagram

may be helpful to keep in mind. We use the fact that direct sums commute with colimits (used in the definition of the direct image sheaf) to simplify notation. We then get sheaves\[\begin{array}{r c l}
\mathcal F^1 & \in & \Shv(A_1), \\
i_*^1\mathcal F^1 \oplus j_*^2 \mathcal F^2 & \in & \Shv(A_2), \\
i_*^2i_*^1\mathcal F^1 \oplus i_*^2j_*^2 \mathcal F^2 \oplus j_*^3 \mathcal F^3 & \in & \Shv(A_3), \\
i_*^3i_*^2i_*^1\mathcal F^1 \oplus i_*^3i_*^2j_*^2 \mathcal F^2 \oplus i_*^3j_*^3 \mathcal F^3 \oplus j_*^4 \mathcal F^4 & \in & \Shv(A_4),
\end{array}\]and finally\[i_*^{N-1\cdots 1}\mathcal F^1 \oplus \left(\bigoplus_{k=2}^{N-1} i_*^{N-1\cdots k}j_*^k \mathcal F^k \right) \oplus j_*^N \mathcal F^N \in \Shv(A_N=X),\]where $i_*^{N-1\cdots k}$ is the composition $i_*^{N-1} \circ i_*^{N-2} \circ \cdots \circ i_*^k$ of direct image functors. Call this last sheaf simply $\mathcal F \in \Shv(X)$. Each $i_*^k$ extends the sheaf by 0 on an ever larger domain, so every summand in $\mathcal F$ is non-zero on exactly one stratum as defined by $f:X\to A$. We now have a functor $\mathcal F:Op(X) \to SC$ defined by\[\mathcal F(U) = \bigsqcup_{k=1}^N S_k \delta_{U,A_K\setminus A_{k-1}},
\hspace{1cm}
\mathcal F_{(P,t)} = \bigsqcup_{k=1}^N S_k \delta_{(P,t),\text{cl}(,A_K\setminus A_{k-1})},\]where $\delta_{U,V}$ is the Kronecker delta that evaluates to the identity if $U\cap V \neq \emptyset$ and zero otherwise.

Remark: The sheaf $\mathcal F$ is $A$-constructible, as $\mathcal F|_{f^{-1}(S)}$ is a constant sheaf evaluating to the simplicial complex $S\in A$. However, if we want the cohomology groups to capture how the simplicial complexes change between strata, then we must use a different approach - all groups die when leaving a stratum because of the extension by zero construction.

References: nLab (article "Simplicial complexes")

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")