The Display Space

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.