CHANGING DATATYPES IN SIMPLICIAL DATABASES
DAVID I. SPIVAK
1. Locales
1.1. Morphisms between locales of subobjects. Suppose that f : X → Y is a
morphism of simplicial sets. Let X = Sub(X) (respectively Y = Sub(Y )) denote
the locale of subobjects of X (resp. Y ). The map f induces an adjunction
/X : F
f −1 : Y o
W
where F(X 0 ) = (Y 0 ⊂ Y |f −1 (Y 0 ) ⊂ X 0 ). Note that f −1 preserves finite meets,
so F : X → Y denotes a morphism of locales. This morphism further induces an
adjunction between Grothendieck topoi
/ Shv(X ) : F
F ∗ : Shv(Y) o
∗
defined as follows for sheaves A ∈ Shv(X ) and B ∈ Shv(Y). For any U ∈ Sub(X)
we take
(1)
F ∗ B(U ) := B(f (U )),
where f (U ) ∈ Sub(Y ) is the image of U in Y . For any V ∈ Sub(Y ) we take
(2)
F∗ A(V ) := A(f −1 (V )),
where f −1 (V ) is the preimage of V in X.
We introduced these ideas using a map of simplicial sets f : X → Y , but in
actuality the definitions of F ∗ and F∗ work whenever F is a open morphism of
locales, as we shall soon see. To get to that point, however, it is best to start all
over, by taking X and Y to be arbitrary locales and considering the adjunction
/X : f
f∗ : Y o
∗
as a morphism F = f∗ of locales. The left adjoint f ∗ induces an adjunction of
Grothendieck topoi denoted
/ Shv(X ) : F
F ∗ : Shv(Y) o
∗
by F∗ , which is given by the formula
(3)
F∗ (A)(V ) := A(f ∗ V ).
The left adjoint F ∗ is easy to understand on representable functors, but in general
is just understandable as a colimit.
We say that F is an open morphism if f ∗ has a left adjoint f! : X → Y satisfying
the “Frobenius identity”
f! (x ∧ f ∗ y) = f! (x) ∧ y
This project was supported in part by the Office of Naval Research.
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DAVID I. SPIVAK
for all x ∈ X and y ∈ Y. In this case, we can do a better job of understanding F ∗ .
Namely, for a sheaf B ∈ Shv(Y) and object U ∈ X , we define
(4)
F ∗ B(U ) := B(f! (U ))
and still
To bring it together, a morphism f : X → Y of simplicial sets always induces an
open morphism of locales. Thus, Equations (1) and (2) becomes Equations (4) and
(3).
2. The current model
Let Loc denote the category of locales (see [?]). Recall that a morphism between
locales is an adjunction (f ∗ , f∗ ) for which f ∗ preserves finite meets; such a morphism
is denoted by the right adjoint part.
Definition 2.0.1. A type specification consists of a pair D = (D, Γ), where D is a
category and Γ ∈ Pre(D) is a presheaf on D. A morphism of type specifications,
denoted
(F, F ] ) : D → D0 ,
consists of a functor F : D0 → D and a map of presheaves F ] : F ∗ (Γ) → Γ0 on D.
Definition 2.0.2. Let D = (D, Γ) denote a type specification. A schema on D
consists of a pair (L, σ), where L is a locale and σ : L → Pre(D) preserves colimits
and finite limits. A morphism of schemas is denoted
F = (f∗ , f ] ) : (L, σ) → (L0 , σ 0 ),
/ L : f is a morphism of locales, and f ] : σ ◦ f ∗ → σ 0 is a natural
where f ∗ : L0 o
∗
transformation of functors L0 → Pre(D).
Definition 2.0.3. Let D = (D, Γ) denote a type specification, and let X = (L, σ)
denote a schema over D. The universal sheaf on X, denoted UX : Lop → Sets, is
the functor which assigns
` 7→ HomPre(D) (σ(`), Γ)
for ` ∈ L.
It is clearly contravariant and takes joins in L to limits in Sets.
Let X 0 = (L0 , σ 0 ). Given a morphism F = (f, f ] ) : X → X 0 of schemas, there is
an induced morphism UF : UX 0 → F∗ (UX ) given as follows for `0 ∈ L0 :
f]
UL0 (`0 ) = HomPre(D) (σ 0 `0 , Γ) −→ HomPre(D) (σf ∗ `0 , Γ) = F∗ (UL )(`0 ).
Definition 2.0.4. Let D = (D, Γ) denote a type specification. A database of type
D is a sequence (X, KX , τX ), where X = (L, σ) is a schema, KX ∈ Shv(L) is a
sheaf of sets on L and τX : KX → UX is a morphism of sheaves over the universal
sheaf UX .
A morphism of databases consists of a pair
(F, F ] ) : (X, KX , τX ) −→ (Y, KY , τY ),
CHANGING DATATYPES IN SIMPLICIAL DATABASES
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where F : X → X 0 is a morphism of schema and F ] : KY → F∗ KX is a morphism
of sheaves on Y , such that the diagram
KY
τY
UF
F]
F∗ KX
commutes.
/ UY
F∗ τX
/ F ∗ UX