Interlude on Precosheaves

In this section we develop the theory of precosheaves to the extend needed for the interleaving distance of Reeb graphs by de Silva, Munch, and Patel (2016) and subsequent sections. Before we get to precosheaves we start with some point-set topological definitions.

  • Definition (Intersection-Base). Let \(X\) be a set. An intersection-base on \(X\) is a collection \(\mathcal{B}\) of subsets of \(X\) that covers \(X\) and such that for any \(U, V \in \mathcal{B}\) we have \(U \cap V \in \mathcal{B}\).

  • Example. Let \(X\) be a topological space, then the topology on \(X\) is an intersection-base on \(X\).

    The set of open intervals of \({\mathbb{R}}\) is an intersection-base on \({\mathbb{R}}\).

  • Definition (Intersection-based space). An intersection-based space is a set \(X\) together with an intersection-base \(\mathcal{B}\) on \(X\).

    For any \(U \in \mathcal{B}\) we say that \(U\) is a distinguished open subset of \(X\).

Similar to topological spaces we can equip subsets of an intersection-based space with an induced intersection-base.

  • Lemma. Let \(X\) be a set and \(\mathcal{B}\) be an intersection base on \(X\). For a subset \(Y \subseteq X\) the family \(\{Y \cap U\}_{u \in \mathcal{B}}\) is an intersection-base on \(Y\).

  • Definition. In the context of the previous lemma we say \(Y\) is a subspace of \(X\).

In addition to subspaces we also define products.

  • Definition (Product). Let \(X\) and \(Y\) be intersection-based spaces with intersection-bases \(\mathcal{B}_X\) respectively \(\mathcal{B}_Y\). We augment \(X \times Y\) with the intersection-base \(\mathcal{B}_X \times \mathcal{B}_Y\) and name this space the product \(X \times Y\) of \(X\) and \(Y\).

Next we define continuous maps between intersection-based spaces.

  • Definition. For two intersection-based spaces \(X\) and \(Y\) a map of sets \(\varphi \colon X \rightarrow Y\) is said to be continuous if for any distinguished open subset \(V \subseteq Y\) it’s preimage \(\varphi^{-1} (V)\) is a distinguished open subset of \(X\).

  • Remark. The previous definition augments the class of intersection-based spaces with the structure of a category that contains the category of topological spaces as a full subcategory.

  • Example. Suppose \(Y\) is a subspace of the intersection-based space \(X\), then the inclusion of \(Y\) into \(X\) is continuous.

  • Lemma. Let \(X\) be a topological space, let \(Y\) be a set, and let \(\mathcal{B}\) be an intersection-base on \(Y\). Then a map of sets \(\varphi \colon X \rightarrow Y\) is continuous as a map between intersection-based spaces if and only if it is continuous with respect to the topology on \(Y\) induced by \(\mathcal{B}\).

  • Remark. The previous lemma implies that the category of topological spaces is a coreflective subcategory of the category of intersection-based spaces.

As we can see intersection-based spaces and topological spaces are very similar and so far intersection-based spaces didn’t provide us with anything new. It is the following definition where intersection-based spaces provide something that their corresponding topological spaces my not provide.

  • Definition. Let \(X\) and \(Y\) be intersection-based spaces with intersection-bases \(\mathcal{B}_X\) respectively \(\mathcal{B}_Y\). A continuous map \(\varphi \colon X \rightarrow Y\) is Galois if for all \(U \in \mathcal{B}_X\) the set \({\{V \in \mathcal{B}_Y ~|~ U \subseteq \varphi^{-1} (V)\}}\) has a minimum which we denote by \(\varphi^{+1} (U)\).

The reason we name such a map \(\varphi\) Galois is that \(\varphi^{+1}\) and \(\varphi^{-1}\) then form a monotone Galois connection \(\varphi^{+1} \dashv \varphi^{-1}\).

  1. Lemma. Let \(\varphi \colon X \rightarrow Y\) and \(\psi \colon Y \rightarrow Z\) be continuous and Galois, then \(\psi \circ \varphi\) is Galois and \((\psi \circ \varphi)^{+1} = \psi^{+1} \circ \varphi^{+1}\).

  • Proof. This follows from a general statement about Galois connections, see for example (Erné et al. 1993, proposition 2).

These are all the point-set topological notions we need, so we can start with precosheaves.

  • Definition (Set-valued Precosheaves). Let \(X\) be an intersection-based space with intersection base \(\mathcal{B}\). A set-valued precosheaf \(F\) on \(X\) is a functor from \(\mathcal{B}\), partially ordered by inclusions, to the category of sets.

The reader may already guess that for an intersection-based space \(X\) we will also augment the class of set-valued precosheaves on \(X\) with the structure of a category.

  • Definition. Let \(F\) and \(G\) be precosheaves on some intersection-based space \(X\). A homomorphism \(\alpha \colon F \rightarrow G\) is a natural transformation from \(F\) to \(G\).

    Composition of homomorphisms is given by composition of natural transformations.

Now let \(\varphi \colon X \rightarrow Y\) be a continuous map between intersection-based spaces. We associate to \(\varphi\) a functor from the category of set-valued precosheaves on \(X\) to the category of precosheaves on \(Y\).

  • Definition (Pushforward). For a precosheaf \(F\) on \(X\) we set \(\varphi_* F := F \circ \varphi^{-1}\) and for a homomorphism \(\alpha \colon F \rightarrow G\) we set \(\varphi_* \alpha := \alpha \circ \varphi^{-1}\). Here we view \(\varphi^{-1}\) as a monotone map from the intersection-base of \(Y\) to the intersection-base of \(X\). We name \(\varphi_* F\) the pushforward of \(F\) by \(\varphi\).

We note that \(\varphi_*\) is a functor from the category of precosheaves on \(X\) to the category of precosheaves on \(Y\). Now let \(\psi \colon Y \rightarrow Z\) be another continuous map between intersection-based spaces.

  • Lemma. We have \((\psi \circ \varphi)_* = \psi_* \circ \varphi_*\).

Now suppose \(\varphi\) and \(\psi\) are Galois. We describe a functor in the other direction.

  • Definition (Pullback). For a precosheaf \(G\) on \(Y\) we set \(\varphi_p G := G \circ \varphi^{+1}\) and for a homomorphism \(\alpha \colon F \rightarrow G\) we set \(\varphi_p \alpha := \alpha \circ \varphi^{+1}\). We name \(\varphi_p G\) the pullback of \(G\) by \(\varphi\).

We note that \(\varphi_p\) is a functor from the category of precosheaves on \(Y\) to the category of precosheaves on \(X\).

  • Lemma. We have \((\psi \circ \varphi)_p = \varphi_p \circ \psi_p\).

  • Proof. This follows from lemma 5.

  • Definition. For all distinguished open subsets \(U \subseteq X\) we have \(U \subseteq (\varphi^{+1} \circ \varphi^{-1})(U)\) and this yields a natural homomorphism \(\eta^{\varphi}_F \colon F \rightarrow \varphi_p \varphi_* F\) for any precosheaf \(F\) on \(X\).

    Conversely we have \((\varphi^{-1} \circ \varphi^{+1})(V) \subseteq V\) for all distinguished open subsets \(V \subseteq Y\). This induces a natural homomorphism \(\varepsilon^{\varphi}_G \colon \varphi_* \varphi_p G \rightarrow G\) for any precosheaf \(G\) on \(Y\).

If \(\varphi\) is not Galois then a functor like \(\varphi_p\) still exists, see for example (Stacks Project Authors 2017, tag 008C), but it is not as easy to define and not as easy to work with. In particular the previous lemma would not hold in this form and such compositions would invoke some subtleties. This is the reason we introduced intersection-based spaces in the first place.

  • Definition (Restriction). For a subspace \(Y\) of an intersection-based space \(X\) whose inclusion is Galois and a precosheaf \(F\) on \(X\) we denote the pullback of \(F\) by the inclusion by \(F |_Y\).