sec-comparator-sketches

12.1 Comparator sketches as descent data

Let \(\mathbb A\) be a small information category, let \(\mathbb R\) be a reference shape, and let

\[ q:\mathbb A\longrightarrow \mathbb R \]

identify distinctions that the comparator must ignore. For a complete target category \(\mathcal D\), precomposition has a right adjoint

\[ q^\ast :[\mathbb R,\mathcal D]\rightleftarrows [\mathbb A,\mathcal D]:\operatorname {Ran}_q. \]

A reference diagram \(r:\mathbb R\to \mathcal D\) is realized on the original information category as \(q^\ast r=r\circ q\).

Definition 12.1 Comparator sketch

A comparator sketch in \(\mathcal D\) is a functor \(q:\mathbb A\to \mathbb R\) together with a replete full subcategory

\[ \mathcal K\hookrightarrow [\mathbb R,\mathcal D] \]

of admissible reference diagrams. Its realized comparator class is the essential image of

\[ q^\ast |_{\mathcal K}:\mathcal K\longrightarrow [\mathbb A,\mathcal D]. \]

The sketch is right-Kan admissible when the adjunction unit

\[ \eta _r:r\longrightarrow \operatorname {Ran}_q q^\ast r \]

is an isomorphism for every \(r\in \mathcal K\).

The functor \(q\) controls variation, while \(\mathcal K\) controls admissibility. These roles are independent. Collapsing all sites may declare a static comparator; preserving an epoch index may declare a block-static one; taking \(q=1_{\mathbb A}\) permits fully dynamic trajectories. Constraints on actions, policies, or resources belong to \(\mathcal K\), not to the quotient shape by itself.

For any trajectory \(c:\mathbb A\to \mathcal D\), the counit

\begin{equation} \varepsilon _c: q^\ast \operatorname {Ran}_q c\longrightarrow c \end{equation}

12.1

compares its canonical right-Kan reconstruction with the trajectory that was actually presented. This is the intrinsic comparison map of the sketch.

Theorem 12.2 Right-Kan representation of comparator sketches

Let \((q,\mathcal K)\) be a right-Kan-admissible comparator sketch and assume the displayed right Kan extensions exist. A trajectory \(c\in [\mathbb A,\mathcal D]\) belongs to the realized comparator class if and only if

\(\operatorname {Ran}_q c\) belongs to \(\mathcal K\), and
the counit \(\varepsilon _c:q^\ast \operatorname {Ran}_qc\to c\) is an isomorphism.

Consequently, the comparator class is characterized internally on \(\mathbb A\), without choosing a coordinate representation of a reference policy.

Proof ▼

Suppose \(c\cong q^\ast r\) for \(r\in \mathcal K\). Right-Kan admissibility gives \(r\cong \operatorname {Ran}_q q^\ast r\), hence \(\operatorname {Ran}_qc\cong r\in \mathcal K\) because \(\mathcal K\) is replete. The triangle identity

\[ \varepsilon _{q^\ast r}\circ q^\ast \eta _r =1_{q^\ast r} \]

and invertibility of \(\eta _r\) show that \(\varepsilon _{q^\ast r}\), and therefore \(\varepsilon _c\), is invertible.

Conversely, if \(\operatorname {Ran}_qc\in \mathcal K\) and \(\varepsilon _c\) is invertible, then it exhibits

\[ c\cong q^\ast (\operatorname {Ran}_qc), \]

so \(c\) lies in the essential image of \(q^\ast |_{\mathcal K}\).

Remark 12.3 The static-comparator case

For a nonempty connected \(\mathbb A\), take \(q=\pi :\mathbb A\to 1\) and \(\mathcal K=\mathcal D\). Then \(q^\ast =\Delta \), and the representation theorem reduces to Proposition 9.18. Thus the familiar static comparator is the coarsest member of a much larger descent semantics.