Sheaves and cosheaves

Let $X$ be a topological space with an open cover $\mathcal U = \{U_i\}$, and category $Op(X)$ of open sets of $X$. Let $C$ be any abelian category, most often groups.

Definition: A presheaf $\mathcal F$ over $X$ is a functor $Op(X)^{op}\to D$, and a sheaf if it satisfies the gluing axiom. A precosheaf $\widehat{\mathcal F}$ over $X$ is a functor $Op(X)\to D$, and a cosheaf if it satisfies the cutting axiom.

The gluing axiom may be interpreted as a colimit condition and the cutting axiom (thanks to Keaton Quinn for suggesting the name) may be interpreted as a limit condition. The components of sheaves and cosheaves are compared in the table below.
\[
\begin{array}{r|c|c}
& \text{sheaf} & \text{cosheaf} \\\hline
&&\\[-5pt]
\text{functoriality} & \begin{array}{r c l}
Op(S)^{op} & \to & D \\
U & \mapsto & \mathcal F(U)\\
(V\hookrightarrow U)^{op} & \mapsto & (\rho_{UV}:\mathcal F(U)\to \mathcal F(V))
\end{array}
&
\begin{array}{r c l}
Op(S) & \to & D \\
U & \mapsto & \widehat{\mathcal F}(U)\\
(V\hookrightarrow U) & \mapsto & (\varepsilon_{VU}:\widehat{\mathcal F}(V)\to \widehat{\mathcal F}(U))
\end{array}
\\&&\\
\text{gluing / cutting} &
\begin{array}{r l}
\text{if} &  s_i|_{U_i\cap U_j}=s_j|_{U_i\cap U_j},\\[5pt]
\text{then} & \begin{array}{c}\exists s\in \mathcal F(U_i\cup U_j) \text{ s.t.}\\ s|_{U_i}=s_i,s|_{U_j}=s_j. \end{array}
\end{array}
&
\begin{array}{r l}
\text{if} & s_i|^{U_i\cup U_j}=s_j|^{U_i\cup U_j},\\[5pt]
\text{then} & \begin{array}{c}\exists s\in \widehat{\mathcal F}(U_i\cap U_j) \text{ s.t.}\\ s|^{U_i}=s_i,s|^{U_j}=s_j. \end{array}
\end{array}
\\&&\\
\text{colimit / limit cond.} &
\mathcal F(U)\tov\cong \displaystyle\varprojlim_{V\subseteq U} \mathcal F(V)
&
\widehat{\mathcal F}(U)\xleftarrow{\hspace{3pt}\cong\hspace{3pt}} \displaystyle\varinjlim_{V\subseteq U} \widehat{\mathcal F}(V)
\end{array}
\]
The maps $\rho_{UV}$ are called restrictions and $\varepsilon_{VU}$ are called extensions. Above, $s_i$ is a (co)section over $U_i$ and $s_j$ is a (co)section over $U_j$. For $s$ a (co)section of $U$ with $V\subset U\subset W$, write $s|_V$ for $\rho_{UV}(s)$ and $s|^W$ for $\varepsilon_{UW}(s)$. The isomorphisms with the colimits and limits are the natural maps from the respective colimit and limit diagrams.

Now we relate sheaves to persistent homology. All cohomology is be taken over a field $k$.

Remark: Suppose we have a finite point sample $P$ and some $t>0$, for which we can construct the nerve $N_{t,P}$, a cellular complex, of the union of balls of radius $t$ around the points of $P$. If $t'<t$, then there is a natural inclusion $N_{t',P}\hookrightarrow N_{t,P}$, which induces a map $H_\ell(N_{t',P})\to H_\ell(N_{t,P})$ on degree $\ell$ homology groups. Define a sheaf $\mathcal F^\ell$ over $\R$ for which
\[
\mathcal F^\ell(U) = H^\ell(N_{\inf(U),P}),
\hspace{1cm}
\mathcal F^\ell_t = H^\ell(N_{t,P}).
\]
This is indeed a sheaf, as $V\subseteq U$ implies that $\inf(U)\leqslant \inf(V)$, giving a natural map $\mathcal F^\ell(U)\to \mathcal F^\ell(V)$. The gluing axiom is also satisfied: assume without loss of generality that $\inf(U_i)\leqslant \inf(U_j)$ and take $s_i\in \mathcal F^\ell(U_i)$, $s_j\in \mathcal F^\ell(U_j)$ with the assumptions as above. Then $\inf(U_i)=\inf(U_i\cup U_j)$ and $\inf(U_j) = \inf(U_i\cap U_j)$, so
\[
\mathcal F^\ell(U_i) = \mathcal F^\ell(U_i\cup U_j),
\hspace{1cm}
\mathcal F^\ell(U_j) = \mathcal F^\ell(U_i\cap U_j),
\]
hence $s_i=s\in \mathcal F^\ell(U_i\cup U_j)$ and $s|_{U_j} = s_i|_{U_j} = s_i|_{U_i\cap U_j} = s_j|_{U_i\cap U_j} = s_j|_{U_j} = s_j$. Therefore sheaves capture all the persistent homology data. Note we do not take the sheaf cohomology of $\mathcal F^\ell$, instead the usual sequence of homology groups is induced by any increasing sequence in $\R$.

References: Bredon (Sheaf theory, Section VI.4), Bott and Tu (Differential forms in algebraic topology, Section 10)