ora-0135

10.6.1 The categorical declaration

Fix a horizon \(T\), dimension \(d\), and field \(k=\mathbb {R}\). Let

\[ \mathcal{P}_T=\{ 0{\lt}1{\lt}\cdots {\lt}T\} \]

be the prefix poset, linearized over \(k\):

\[ \mathcal{P}_T(s,t) = \begin{cases} k,& s\le t,\\ 0,& s{\gt}t. \end{cases} \]

Let \(\mathcal{O}_T\) be the discrete \(k\)-linear category on \(\{ 1,\ldots ,T\} \), and let \(J:\mathcal{O}_T\to \mathcal{P}_T\) send \(s\) to \(s\).

The sufficient-statistic carrier for a quadratic loss is

\[ E_d=\operatorname {Sym}_d(k)\oplus k^d\oplus k. \]

For

\[ \ell _s(x)=\frac12x^\mathsf TQ_sx+b_s^\mathsf Tx+c_s, \]

the revealed statistic is \(e_s=(Q_s,b_s,c_s)\in E_d\). Define the carrier functor \(F:\mathcal{O}_T\to \mathbf{Vect}_k\) by \(F(s)=E_d\). Since \(\mathcal{O}_T\) is discrete, the observations \((e_s)\) form a freely registered local family.

Define the additive realization

\[ \sigma _t: \bigoplus _{s\le t}E_d\longrightarrow E_d, \qquad \sigma _t((z_s)_{s\le t})=\sum _{s\le t}z_s. \]

The distinction between the Kan object and \(\sigma _t\) is not cosmetic: the coproduct in vector spaces is the formal direct sum, not the numerical sum of its entries.