lin-0083

6.1 From functorial databases to tangent instances

Let \(\mathcal S\) be a small category presenting a database schema and let \(\mathcal C\) be a semantic category. The category of \(\mathcal C\)-valued instances is

\[ \mathbf{Db}_{\mathcal C}(\mathcal S) = [\mathcal S,\mathcal C]. \]

An object is a functor \(I:\mathcal S\to \mathcal C\); a morphism is a natural transformation between instances. The familiar functorial database model takes \(\mathcal C=\mathbf{Set}\). For infinitesimal data, however, the semantic category must carry nontrivial tangent structure.

Assume that

\[ (\mathcal C,T,p,0,+,\ell ,c) \]

is a tangent category in the sense of Cockett and Cruttwell [ Cockett and Cruttwell , 2014b ] . Define an endofunctor on instances by first taking an instance morphism

\[ \alpha :I\Rightarrow J \]

to be a natural transformation between two instance functors \(I,J:\mathcal S\to \mathcal C\), and then setting

\[ T_{\mathcal S}I = T\circ I, \qquad T_{\mathcal S}\alpha = T\alpha . \]

Here \(T\alpha :T\circ I\Rightarrow T\circ J\) denotes postcomposition of the natural transformation by \(T\); componentwise,

\[ (T_{\mathcal S}\alpha )_s =T(\alpha _s):T(I(s))\longrightarrow T(J(s)). \]

Thus

\[ (T_{\mathcal S}I)(s)=T(I(s)), \qquad (T_{\mathcal S}I)(f)=T(I(f)). \]

Proposition 6.1 Pointwise tangent database

If \(\mathcal C\) is a tangent category and \(\mathcal S\) is small, the functor category \([\mathcal S,\mathcal C]\) inherits tangent structure pointwise. Its tangent functor is \(T_{\mathcal S}\), and the structure maps \(p,0,+,\ell ,c\) are defined componentwise.

Proof ▼

Limits in a functor category are computed pointwise whenever the corresponding limits exist in \(\mathcal C\). For every object \(s\) of the schema, use the tangent structure map at \(I(s)\). Naturality in \(s\) follows from naturality of the structure maps in \(\mathcal C\). Every tangent category axiom then holds at each component because it holds in \(\mathcal C\). The pullbacks selected by the tangent axioms are likewise computed componentwise.

This proposition gives a precise version of the statement that a database category has a tangent functor. The tangent structure is inherited from the semantic codomain; it is not generated by a bare schema category.

Boundary

For \(\mathbf{Set}\)-valued instances, the pointwise construction supplies no useful differential geometry unless \(\mathbf{Set}\) is equipped with some additional nontrivial semantics. One must instead use smooth, synthetic, differential, or smoothly parameterized instance values. A finite table can remain as a discrete provenance layer while its numerical attributes live in a tangent category.

A useful hybrid semantics is a product category such as

\[ \mathbf{Set}_{\mathrm{disc}}\times \mathbf{Smooth}, \qquad T(A,M)=(A,TM). \]

Keys, source identifiers, and provenance remain discrete; state spaces, parameters, fields, and residuals carry the nontrivial tangent structure. More generally, one may work with bundles or smooth families of instances over a parameter object \(\Theta \).