We summarize some concluding results from (Funk 1995). We assume that $$M$$ is a complete metric space.

• Definition. For a locally connected topological space $$X$$, we denote by $$\Lambda(X)$$ it’s set of connected components. Given an open subset $$U \subset X$$ we denote by $$\Lambda(U)$$ the set of connected components of $$U$$, where we augment $$U$$ with the subspace topology. This defines a cosheaf on $$X$$ with values in the category of sets. Given a continuous map $$f \colon X \rightarrow M$$ we denote by $$\lambda_f$$ the pushforward $$f_* \Lambda$$ and obtain the functor $$\lambda$$ from the category of locally connected topological spaces over $$M$$ to the category of set-valued cosheaves on $$M$$.

For a set-valued (pre)cosheaf Funk (1995) provides a construction similar to the étalé space of a (pre)sheaf.

• Definition. Given a cosheaf $$D$$ on $$M$$ we define it’s display space $${\operatorname{dis}}D$$ as the disjoint union of costalks $$S_p := \big\{ x \in \prod_{p \in U \in \mathcal{O}(M)} D U ~ \big| ~ D_{U, V} \big( \pi_U (x) \big) = \pi_V (x) ~~ \text{for all} ~~ U \subseteq V \big\}$$ over all $$p \in M$$ where $$\mathcal{O} (M)$$ is the set of open subsets of $$M$$. Further we define $$\gamma_D \colon {\operatorname{dis}}D \rightarrow M$$ by $$\gamma_D (x) = p$$ for all $$x \in S_p$$. To specify a topology on $${\operatorname{dis}}D$$ we provide as a basis $$\{(U, b)\}_{U \in \mathcal{O}(M),~ b \in D U}$$ where $$(U, b) := \{x \in \gamma_D^{-1} (U) ~|~ \pi_U (x) = b\}$$ for $$U \in \mathcal{O}(M)$$ and $$b \in D U$$.

By Funk (1995 Theorem 6.1) $${\operatorname{dis}}D$$ is locally connected for any cosheaf $$D$$ on $$M$$.

• Assumption. From this point on we assume all topological spaces to be locally connected.

We continue to specify a natural transformation $$\eta$$ from $${\operatorname{id}}$$ to $$\gamma \circ \lambda$$.

• Definition. Given a continuous map $$f \colon X \rightarrow M$$, $$x \in X$$ and $$U \in \mathcal{O}(M)$$ with $$f(x) \in U$$ let $$[x]_U \in \lambda_f (U) = \Lambda (f^{-1} (U))$$ be the connected component of $$x$$. Now we define $$\eta_f \colon X \rightarrow {\operatorname{dis}}\lambda_f, x \mapsto ([x]_U)_{f(x) \in U \in \mathcal{O}(M)}$$.

• Lemma. With $$\eta_f$$ defined as above we have $$(\gamma \circ \lambda)_f \circ \eta_f = f$$.

• Proof. Given $$x \in X$$ we have by the definition of $$\eta_f$$ that $$\eta_f (x) \in S_{f(x)}$$, hence $$(\gamma \circ \lambda)_f (\eta_f (x)) = f(x)$$ by the definition of $$(\gamma \circ \lambda)_f$$.

• Lemma. The map $$\eta_f$$ as defined above is continuous.

• Proof. Given $$U \in \mathcal{O}(M)$$ and $$b \in \lambda_f (U) = \Lambda (f^{-1} (U))$$ we need to show that $$\eta_f^{-1} (\{x \in (\gamma \circ \lambda)_f^{-1} (U) ~|~ \pi_U (x) = b\})$$ is open. To do so we will show that $$\eta_f^{-1} (\{x \in (\gamma \circ \lambda)_f^{-1} (U) ~|~ \pi_U (x) = b\}) = b$$ which is open, since $$X$$ is locally connected. Suppose that $$x \in X$$ is such that $$\eta_f (x) \in (\gamma \circ \lambda)_f^{-1} (U)$$ and $$\pi_U (\eta_f (x)) = b$$, then $$x \in \eta_f^{-1} ((\gamma \circ \lambda)_f^{-1} (U)) = f^{-1} (U)$$ by the previous lemma. Further we have $$b = \pi_U (\eta_f (x)) = \pi_U (([x]_V)_{f(x) \in V \in \mathcal{O}(M)} = [x]_U$$, hence $$x \in b$$. The converse follows from a similar argument.

• Definition. Let $$f \colon X \rightarrow M$$ be continuous, then $$f$$ is a cosheaf space over $$M$$ if $$\eta_f$$ is a homeomorphism.

By (Funk 1995, Theorem 5.9 and Remark 5.10) $$\lambda$$ and $$\gamma$$ form a pair of adjoint functors $$\lambda \dashv \gamma$$ with unit $$\eta$$. Further the counit $$\varepsilon$$ for this adjunction is a natural isomorphism by (Funk 1995, Theorem 6.1). We summarize this as a

1. Theorem. $$\lambda$$ and $$\gamma$$ form a pair of adjoint functors $$\lambda \dashv \gamma$$ with unit $$\eta$$ and whose counit $$\varepsilon$$ is an isomorphism.

• Corollary. The category of cosheaves on $$M$$ is equivalent to the reflective subcategory of cosheaf spaces over $$M$$.

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

• Remark. Beyond the above Funk (1995 Theorem 5.17) provides a topological characterization of cosheaf spaces which hasn’t been mentioned here.