(i_x)_*F(x))\cong \prod_x \Hom(F,\prod_x(i_x)_*F(x)) \cong \prod_x \Hom(F(x),F(x)) \ni \prod_x \id_{F(x)}.
	\]
	We encourage the reader to describe this map is explicitly, by seeing how a single $\id_{F(x)}$ traces through this adjunction, which we'll call $\iota_x\in \Hom(F,(i_x)_*F(x))$.
	
	Similarly, for a cosheaf $\cF:X^{op}\to\Vect$ on an Alexandrov space we could have built a projective surjection by taking
	\[
	 \hat{P}_0:=\bigoplus_{x} (i_x)_*\cF(x) \to \cF \to 0
	\]
	where the map $\pi_0:P_0\to \cF$ is gotten through the corresponding adjunction for cosheaves and using the contravariance of $\Hom$ in the first slot
	\[
		\Hom(\bigoplus_{x} (i_x)_*\cF(x),\cF)\cong \prod \Hom((i_x)_*\cF(x),\cF)\cong \prod \Hom(\cF(x),\cF(x)).
	\]
\end{proof}

\begin{cor}
	Every sheaf $F:X\to\Vect$ has an injective resolution. Dually, every cosheaf $\cF:X^{op}\to\Vect$ has a projective resolution.
\end{cor}
\begin{proof}
	Since cokernels exist in the category of sheaves by taking element-by-element quotients and by iteratively applying the claim, we obtain an injective resolution of $F$:
	\[
	 \xymatrix{F \ar[r]^{\iota^0} & I^0 \ar[rd]_{\pi_0} \ar[rr]^{\iota^1=j_1\pi_0} & & I^1 \ar[rr]^{\iota^2=j_2\pi_1} \ar[rd]_{\pi_1} & & I^2 &\cdots \\
		      & & \cok(\iota^0) \ar[ru]_{j_1} & & \cok(\iota^1) \ar[ru]_{j_2} & &\cdots }
	\]
	
	Iterating the analogous process for projective cosheaves, replacing kernels where one sees cokernels above, one obtains an exact sequence of cosheaves called the \textbf{projective resolution} of $\cF$:
	\[
	 \cdots \hat{P}_2 \to \hat{P}_1 \to \hat{P}_0 \to \cF \to 0.
	\]
\end{proof}

These exact sequences can be used to replace $F$ or $\cF$ in a suitable sense, defined by the derived category. Before moving onto that discussion, we note one interesting point.

\begin{prop}
 The length of injective resolution of any sheaf $F\in\Shv(X)$ is bounded by the length of longest chain in the poset. In particular for $X$ a cell complex, it is bounded by the dimension.
\end{prop}
\begin{proof}
 Pick a maximal ordered subset in $X$ and consider its top element, say $x'$, then $I^0(x')=F(x')$ since nothing is larger than $x'$. The cokernel sheaf of $\iota^0$ evaluated on $x'$ is then $\cok(\id:F(x')\to F(x'))=0$. So for any maximally ordered chain in $X$, $I^1$ is zero on the top-most element. Arguing inductively finishes the proof.
\end{proof}

\subsection{The Derived Category and Homotopy Theory of Chain Complexes}
\label{subsec:htpy_chain}

The purpose of the derived category is to replace the category of sheaves with a category of complexes where certain operations are more natural. We have already shown that one can replace a sheaf by its injective resolution and a cosheaf by its projective resolution. This will define our derived replacement on the level of objects, but we have not yet shown how a map of sheaves or cosheaves induces a map on the level of resolutions. 

If $\phi:\cF\to\cG$ is a map of cosheaves, then it can be checked from the universal properties of projective objects, that this induces a map of complexes
\[
	\xymatrix{ \cdots & \hat{P}_2 \ar[r] \ar@{.>}[d]^{\phi_2} & \hat{P}_1 \ar[r] \ar@{.>}[d]^{\phi_1} & \hat{P}_0 \ar[r] \ar@{.>}[d]^{\phi_0} & \cF \ar[d]^{\phi} \ar[r] & 0\\ \cdots & \hat{Q}_2 \ar[r] & \hat{Q}_1 \ar[r] & \hat{Q}_0 \ar[r] & \cG \ar[r] & 0}
\]
where all the squares in sight commute. For a hint on how to see this, consider the composite map $\hat{P}_0\to\cF\to\cG$ and let $\cG=A$ and $B=\hat{Q}_0$ in the definition of the universal property defining a projective object. This induces our first map $\hat{P}_0\to\hat{Q}_0$. To get the next, all important step, one must recognize that having maps from $\hat{P}_0\to \hat{Q}_0$ and $\cF\to\cG$ induces maps between the kernels of the map $\hat{P}_0\to\cF$ and $\hat{Q}_0\to\cG$. Since $\hat{Q}_1$ surjects onto the kernel of the latter map repeating the initial argument provides a map from $\hat{P}_1$ to $\hat{Q}_1$. This shows that the projective replacement of cosheaves is functorial.

Aside from functoriality, there is one more snag that needs to be mentioned: For a sheaf or a cosheaf it is possible that the choice of injective or projective resolution is not unique. If one really wants to use these as replacements for the original sheaf or cosheaf, there must be a strong relationship between these two complexes. This is best seen by specializing the functoriality discussion above to the case $\phi=\id$.
\[
	\xymatrix{ \cdots & \hat{P}_2 \ar[r] \ar@{.>}[d]^{\phi_2} & \hat{P}_1 \ar[r] \ar@{.>}[d]^{\phi_1} & \hat{P}_0 \ar[r] \ar@{.>}[d]^{\phi_0} & \cF \ar[d]^{\id} \ar[r] & 0\\
	\cdots & \hat{Q}_2 \ar[r] & \hat{Q}_1 \ar[r] & \hat{Q}_0 \ar[r] & \cF \ar[r] & 0}
\]
The resulting map of complexes need not be a term-by-term isomorphism with all squares in sight commuting, but rather a more general notion must be substituted, namely the definition of chain homotopy. Before giving that, let us give an example.

\begin{ex}[Non-Unique Projective Resolutions]
	Let us work again over our test space of the closed unit interval $X=[0,1]$ stratified as $x=0$, $y=1$ and $a=(0,1)$. The constant cosheaf $\hat{k}_X$ is then modeled as
	\[
		\xymatrix{ & \ar[ld]_1 k \ar[rd]^1 & \\ k & & k}
	\]
	Revisiting the definition of the elementary projective cosheaves, there is one obvious projective resolution because the constant cosheaf on this stratification of the unit interval is already projective, so we have the identity map
	\[
		\hat{P}_{\bullet}: \qquad [\hat{a}] \to \hat{k}_X.
	\]
	On the other hand, following blindly the prescription provided for computing the projective resolution of an arbitrary cellular cosheaf would have lead us to the following ``canonical'' resolution:
	\[
		\hat{Q}_{\bullet}: \qquad [\hat{x}]\oplus [\hat{y}]\to [\hat{a}]\oplus [\hat{x}]\oplus [\hat{y}] \to \hat{k}_X
	\]
\end{ex}

\begin{defn}[Chain Homotopy]
	Suppose $(A^{\bullet},d_A)$ and $(B^{\bullet},d_B)$ are two (cohomological) chain complexes and $\phi^{\bullet}$ and $\psi^{\bullet}$ are two chain maps, then \textbf{a chain homotopy} $h