#### Ascending Cosheaves

In addition to the space $${\mathbb{R}}$$ we consider the reals augmented with a coarser topology.

• Definition. Let $$\bar{{\mathbb{R}}}$$ be the topological space $$({\mathbb{R}}, \{(-\infty, r)\}_{-\infty \leq r \leq \infty})$$, then we have the continuous map $${\operatorname{id}}\colon {\mathbb{R}}\rightarrow \bar{{\mathbb{R}}}, x \mapsto x$$.

We can pushforward cosheaves on $${\mathbb{R}}$$ to cosheaves on $$\bar{{\mathbb{R}}}$$ via $${\operatorname{id}}$$.

• Definition. Given a cosheaf $$F$$ on $${\mathbb{R}}$$ and $$-\infty \leq r \leq \infty$$ we define $${\operatorname{id}}_* F ((-\infty, r)) = F ((-\infty, r))$$.

Similar to defining the pullback for sheaves, we take two steps to define the pullback of a cosheaf via $${\operatorname{id}}$$.

• Definition. Given a cosheaf $$F$$ on $$\bar{{\mathbb{R}}}$$ and an open subset $$U \subseteq {\mathbb{R}}$$ we define $${\operatorname{id}}^+ F (U) := F ((-\infty, \sup U))$$.

With this definition $${\operatorname{id}}^+ F$$ is merely a precosheaf for all we know. Yet we have the following.

• Lemma. Given a cosheaf $$F$$ on $$\bar{{\mathbb{R}}}$$ the precosheaf $${\operatorname{id}}^+ F$$ is a cosheaf on the poset of open intervals.

• Proof. Let $$(a, b) = \bigcup_{i \in I} (a_i, b_i)$$. Without loss of generality we assume that $$I$$ has a linear order such that $$b_i \leq b_j$$ for all $$i \leq j$$ and $$a_i \leq a_j$$ for all $$i \leq j$$ with $$b_i = b_j$$. Since $$b = \sup_{i \in I} \sup (a_i, b_i) = \sup_{i \in I} b_i$$ and since $$F$$ is a cosheaf we have the coequalizer diagram $\xymatrix{ \coprod_{i < j} F((-\infty, b_i)) \ar@/^3pt/[r]^{\sigma} \ar@/_3pt/[r]_{\sigma'} & \coprod_{i \in I} F((-\infty, b_i)) \ar[r] & F((-\infty, b)) }$ where $$\sigma_{i,j}$$ maps $$F((-\infty, b_i))$$ identical to $$F((-\infty, b_i))$$ and $$\sigma'_{i,j}$$ maps $$F((-\infty, b_i))$$ to $$F((-\infty, b_j))$$ via the induced inclusion. Now suppose we have $$i < j$$ such that $$b_i \leq a_j$$, then since $$[b_i, a_j]$$ is compact we can find $$i < i_1 < ... < i_k$$ such that $$[b_i, a_j] \subseteq \bigcup_{l=1}^k (a_{i_l}, b_{i_l})$$ and such that this cover is minimal. If $$i_k < j$$ we set $$\tau = \sigma_{i_k, j}$$, $$\tau' = \sigma'_{i_k, j}$$ and if $$j < i_k$$ we set $$\tau = \sigma_{j, i_k}$$, $$\tau' = \sigma'_{j, i_k}$$. With this we have $$\sigma'_{i, j} \circ \sigma_{i, j}^{-1} = \tau' \circ \tau^{-1} \circ \sigma'_{i_{k-1}, i_k} \circ \sigma_{i_{k-1}, i_k}^{-1} \circ ... \circ \sigma'_{i_1, i_2} \circ \sigma_{i_1, i_2}^{-1} \circ \sigma'_{i, i_1} \circ \sigma_{i, i_1}^{-1}$$ and yet at the same time $$b_j > a_{i_k}$$, $$b_{i_k} > a_j$$, $$b_{i_{k-1}} > a_{i_k}$$, …, $$b_{i_1} > a_{i_2}$$, and $$b_i > a_{i_1}$$, since $$[b_i, a_j]$$ is connected and the cover is minimal. Thus we may omit all terms from the leftmost coproduct in the above diagram where $$b_i \leq a_j$$ without loosing the property of it being a coequalizer diagram. Now for any $$i < j$$ such that $$b_i > a_j$$ we have $${\operatorname{id}}^+ F((a_i, b_i) \cap (a_j, b_j)) = F((-\infty, b_i))$$, hence we may replace the corresponding term in the leftmost coproduct by $${\operatorname{id}}^+ F((a_i, b_i) \cap (a_j, b_j))$$. Similarly we may replace the terms in the middle and the term on the right to arrive at a coequalizer diagram of the form $\xymatrix@C-8pt{ \coprod_{i < j,\, b_i > a_j} {\operatorname{id}}^+ F((a_i, b_i) \cap (a_j, b_j)) \ar@/^3pt/[r] \ar@/_3pt/[r] & \coprod_{i \in I} {\operatorname{id}}^+ F((a_i, b_i)) \ar[r] & \, {\operatorname{id}}^+ F((a, b)). }$ Now for $$i < j$$ such that $$b_i \leq a_j$$ we have $${\operatorname{id}}^+ F((a_i, b_i) \cap (a_j, b_j)) = {\operatorname{id}}^+ F(\emptyset) = F(\emptyset) = \emptyset$$ which does not contribute to the coequalizer and this implies the claim.

• Definition. Given a cosheaf $$F$$ on $$\bar{{\mathbb{R}}}$$ and an open subset $$U \subseteq {\mathbb{R}}$$ we define $${\operatorname{id}}^{-1} F (U) := \varinjlim_{(a, b) \subseteq U} {\operatorname{id}}^+ F ((a, b))$$.

By the previous lemma $${\operatorname{id}}^{-1} F$$ as defined above is a cosheaf.

• Remark. $${\operatorname{id}}^{-1} F$$ as defined above is isomorphic to the cosheafification of $${\operatorname{id}}^+ F$$ or the cosheaf associated to $${\operatorname{id}}^+ F$$, see for example (Funk 1995, Theorem 6.3 and Remark 6.4).

• Definition. Given a cosheaf $$F$$ on $${\mathbb{R}}$$ we define a homomorpism $$\eta'_F$$ of cosheaves from $$F$$ to $${\operatorname{id}}^{-1} {\operatorname{id}}_* F$$. Since both are cosheaves it suffices to define $$\eta'_F$$ on open intervals. So for $$-\infty \leq a < b \leq \infty$$ we define $$\eta'_F$$ from $$F((a, b))$$ to $${\operatorname{id}}^{-1} {\operatorname{id}}_* F ((a, b)) = {\operatorname{id}}^+ {\operatorname{id}}_* F ((a, b)) = F((-\infty, b))$$ to be the map induced by the inclusion $$(a, b) \subseteq (-\infty, b)$$.

• Definition. Let $$F$$ be a cosheaf on $${\mathbb{R}}$$, then $$F$$ is ascending if $$\eta'_F$$ is an isomorphism.

1. Proposition. $${\operatorname{id}}_*$$ and $${\operatorname{id}}^{-1}$$ form a pair of adjoint functors $${\operatorname{id}}_* \dashv {\operatorname{id}}^{-1}$$ with unit $$\eta'$$ and whose counit $$\varepsilon'$$ is an isomorphism.

• Proof. Let $$F$$ be a cosheaf on $${\mathbb{R}}$$, let $$G$$ be a cosheaf on $$\bar{{\mathbb{R}}}$$, and let $$g$$ be a homomorphism from $$F$$ to $${\operatorname{id}}^{-1} G$$. Now suppose we have a morphism $$f$$ from $${\operatorname{id}}_* F$$ to $$G$$ such that $$({\operatorname{id}}^{-1} f) \circ \eta'_F = g$$, then for any $$r \in {\mathbb{R}}$$ we have $$g_{(-\infty, r)} = \big(({\operatorname{id}}^{-1} f) \circ \eta'_G \big)_{(-\infty, r)} = f_{(-\infty, r)}$$ and this determines $$f$$. Now suppose $$f$$ is defined by $$g_{(-\infty, r)} = f_{(-\infty, r)}$$ for any $$r \in {\mathbb{R}}$$ and we have $$-\infty \leq a < b \leq \infty$$, then $$g_{(a, b)}$$ is the same as $$g_{(-\infty, b)}$$ pre-composed with the map induced by inclusion from $$F((a, b))$$ to $$F((-\infty, b))$$ by naturality. But this is the same as $$\big(({\operatorname{id}}^{-1} f) \circ \eta'_G \big)_{(a, b)}$$ by definition of $$f$$, hence $$g$$ and $$({\operatorname{id}}^{-1} f) \circ \eta'_G$$ agree on a basis of $${\mathbb{R}}$$.

By the above argument $$\varepsilon'_G$$ is equal to $${\operatorname{id}}_{\, {\operatorname{id}}^{-1} G}$$ when restricted to $${\operatorname{id}}_* {\operatorname{id}}^{-1} G ((-\infty, r)) = {\operatorname{id}}^{-1} G ((-\infty, r)) = G ((-\infty, r))$$, hence $$\varepsilon'_G$$ is an isomorphism.

• Corollary. The category of cosheaves on $$\bar{{\mathbb{R}}}$$ is equivalent to the reflective subcategory of ascending cosheaves on $${\mathbb{R}}$$.

• Proof. This follows with (Gabriel and Zisman 1967, Proposition 1.3 or http://ncatlab.org/nlab/show/reflective+subcategory#characterizations).