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