RULED SPACES
DAVID I. SPIVAK
We begin by defining the category of rulers. Given a topological space X, let
Op(X) denote its set of open sets (or category of open inclusions).
Definition 0.1. A ruler is a pair (R, J), where R is a topological space, and
J = {j1 , . . . , jn } ⊂ Op(R) a finite set of open subsets of R, and these open sets
cover R. If J is understood, we may write R to denote the ruler (R, J).
A morphism of rulers f : (R, J) → (S, K) consists of a continuous map f : R → S
such that, for each open set j ∈ J there exists a k ∈ K such that j ⊂ f −1 (k).
With these definitions, we have defined the category of rulers, denoted Rul.
Example 0.2.
(1) A finite set R canonically has the structure of a ruler (R, R).
The initial and the terminal objects of the category of rulers are of this
form.
(2) If R is a topological space, then the single open set {R} ⊂ Op(R) is a cover.
We denote by R̄ the ruler (R, {R}).
(3) If (R, J) is a ruler and j ∈ J is an open set, then j̄ := (j, {j}) is a ruler,
and there is a canonical morphism j̄ → (R, J) in the category of rulers.
(4) More generally, if (R, J) is a ruler and f : S → R is a subspace, then one
can take K ⊂ Op(S) to be the set of preimages f −1 (j) for j ∈ J. Then f
gives a morphism (S, K) → (R, J).
(5) Consider a ruler in the usual sense: a closed interval in R together with
n-many evenly spaced markings (of a distance from one another). Take R
to be the topological space given by the interval, and take J to be a cover
given by -balls around the marked points. This is a ruler. A morphism of
rulers which leaves the underlying space fixed is simply a refinement of the
cover.
Lemma 0.3. The category Rul of rulers is closed under finite limits.
Proof. It suffices to check that Rul is closed under equalizers and finite products.
Each is easy to check.
Definition 0.4. A subcategory R ⊂ Rul is called a culture if it is a full subcategory
and if, for every ruler (R, J) ∈ R and open set j ∈ J, the ruler j̄ is also an object
in R.
Given a culture R, an interpretation of R is a functor F : R → Sets that preserves finite limits. A morphism of interpretations is just a natural transformation
of functors.
Note that if F : R → Sets is an interpretation, R = (R, J) ∈ R is a ruler, and
j ∈ J is an element of the distinguished cover, then the natural map F (j̄) → F (R)
is a monomorphism, because F preserves finite limits and hence monomorphisms.
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DAVID I. SPIVAK
Lemma 0.5. Let R be a culture. The category of interpretations of R is closed
under taking filtered colimits.
Proof. If S is any category, then a filtered colimit of functors S → Sets, each of
which preserves finite limits, will preserve finite limits as well.
Definition 0.6. Let R be a culture. A preschema on R is a pair (X, OX ), where
X is a topological space and OX is a sheaf of interpretations on R. Given an open
set U ⊂ X and a ruler R ∈ R, we denote by OX (U, R) the set OX (U )(R).
A morphism of preschemas is written
(f, f ] ) : (X, OX ) → (Y, OY ),
where f : X → Y is a map of topological spaces and f ] : f ∗ OY → OX is a morphism
of sheaves.
Let (X, OX ) be a preschema and U ⊂ X an open subset. For each ruler R =
(R, J) ∈ R, we can pretend that the set OX (U, R) is a set of continuous maps
U → R. To make this work, we need a locality condition.