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Spaces from Local Models

Table of Contents

We are often faced with a notion of a local model of spaces that we then glue together to a notion of spaces. This node gives a general framework that abstracts this procedure from the known examples of schemes (from affine schemes to schemes) and manifolds (from open subsets of some \( \mathbb R ^n \) to \( n \)-dimensional manifolds). More precisely we have a category of local models and want to form a category of spaces where every space can be covered by local models and is uniquely (up to isomorphism) determined by such a covering (is the colimit over the cover in the category of spaces).

1. Category of Local Models

A category of local models is in its minimal form a normal and subcanonical site. Let us explain why:

  1. We have to specify a collection of local models and whats maps there are between them, hence a category of local models.
  2. Since we want to glue local models to get spaces, we have to specify how to get a local model from gluing local models, hence we have to specify a coverage.
  3. We wish that all local models are spaces themselves, hence we need a normal and subcanonical coverage. (Subcanonical implies that every local model is a sheaf. Normal implies that every local model is locally representable.)

These requirements are enough to define locally representable sheaves and prove that every local model is a locally representable sheaf. But they are not enough to prove the desired colimit property.

  1. If we want the colimit property, then additionally our site must be cartesian.

Therefore the best or standard setup for our construction is a standard site. Indeed the category of affine schemes with the Zariski coverage and the category of open subsets of \( \mathbb R ^n \) with the coverage induced by open embeddings are standard sites.

1.1. TODO Maybe I should check this once …

2. Construction of Category of Spaces

Let \( \mathcal C \) be a normal and subcanonical site (a category of local models). Then we can construct the category of spaces as follows:

  1. Form the category of sheaves \( \Sh(\mathcal C) \), a sheaf is a generalized space (see here). But a sheaf can’t be covered by local models in general.
  2. Define a coverage on the category of sheaves.
  3. A space is locally representable sheaf, ie a sheaf that can be covered using local models.

Let \( \mathcal C \) be a subcanonical site. Then the induced coverage on \( \Sh(\mathcal C) \) is given by: a family \( (f_i \colon X_i \rightarrow X) \) is a covering family iff for every local model (representable sheaf) \( U \) and morphism \( g \colon U \rightarrow X \)

  • for every \( i \in I \), the pullback \( g^* f_i \colon g^*U_i \rightarrow U \) exists and \( U_i \) is a local model (representable),
  • \( (g^* f_i \colon U_i \rightarrow U) _{i \in I} \) is a covering family in \( \mathcal C \).

As always (with presheaves or sheaves of a subcanonical site) the idea is to probe with representables if a property holds.

Let \( \mathcal C \) be a cartesian and subcanonical site. Then the Yoneda embedding sends a cover in \( \mathcal C \) to a cover in \( \Sh(\mathcal C) \).

Proof.

2.1. QUESTION Which properties are inherited from the site \( \mathcal C \) to \( \Sh(\mathcal C) \)?

Let \( \mathcal C \) be subcanonical site. A sheaf \( X \in \Sh(\mathcal C) \) is locally representable iff there exists a covering family \( (f_i \colon U_i \rightarrow X) \) in the induced coverage on \( \Sh(\mathcal C) \) such that for every \( i \in I \) the sheaf \( U_i \) is representable.

Let \( \mathcal C \) be a standard site (category of local models). Then the induced category of spaces \( \Sp(\mathcal C) \) is the full subcategory of \( \Sh(\mathcal C)\) of locally representable sheaves.

Let \( \mathcal C \) be a normal and subcanonical site. Then:

  1. Every local model (representable sheaf) is a locally representable sheaf.
  2. The Yoneda embedding factors \( \mathcal C \rightarrow \Sp(\mathcal C) \rightarrow \Sh(\mathcal C) \).
Proof.

Let \( \mathcal C \) be a cartesian and subcanonical site. Let \( (f_i \colon U_i \rightarrow X) _{i \in I} \) be a cover of \( X \) by local models in \( \Sh(\mathcal C) \). The index category \( J(I) \) induced by the cover consists of

  • an object \( \{i, j\} \) for every \( i, j \in I \) (possibly \( i = j \)).
  • a morphism \( \{i, j\} \rightarrow \{k,l\} \) iff \( \{k,l\} \subseteq \{i, j\} \).

The by the cover induced diagram is the functor \(F((f_i) _{i \in I}) \colon J(I) \rightarrow \mathcal C \) where

  • \( \{i,j\} \mapsto f^*_i U_j (\cong f^*_j U_i) \),
  • \((\{i,j\} \rightarrow \{i\}) \mapsto f^*_i f_j \).

The by the cover induced colimit is \( \colim F((f_i) _{i \in I}\). Note that we can interpret this colimit in \( \mathcal C \) or \( \Sh(\mathcal C) \).

In words: the induced colimit of a cover (of local models) consists of all local models appearing in the cover pairwise connected by the corresponding pullback.

A locally representable sheaf is the colimit induced by a covering (of local models) (in the category of presheaves, then also in sheaves and locally representable sheaves). Let \( \mathcal C \) be a standard site. Then:

  1. If \( (f_i \colon U_i \rightarrow U) _{i \in I}\) is a cover in \( \mathcal C \), then \( U \) is the induced colimit, \( \colim F((f_i) _{i \in I} \cong U \) in \( \Sh(\mathcal C)\) (hence also in \( \Sp(\mathcal C)\)).
  2. If \( (f_i \colon U_i \rightarrow X _{i \in I} \) is a cover in \( \Sh(\mathcal C) \) of \( X \) using local models, then \( X \) is the induced colimit in \( \Sh(\mathcal C)\) (hence also in \( \Sp(\mathcal C)\)).
Proof. \begin{equation*} X(V) \cong \PSh(\mathcal C)(V,X). \end{equation*} \begin{equation*} g \mapsto \Glue (h_i \circ f_i^*g) _{i \in I}. \end{equation*} \begin{align*} Y(s)(h_i \circ f_i^*g) = & h_i \circ f_i^*g \circ s\\ = & h \end{align*}

This proof is very instructive!

Author: Frederik Gebert

Created: 2025-02-10 Mon 21:45