Example [efr-BNB2]
  1. Let \mathcal {C} = \mathsf {Optic}(\mathsf {Set}). There is a selection function \mathrm {argmax}_X \in \mathbb {S}({\mathbb {R} \choose X}) defined by \mathrm {argmax}(x,f) if and only if x is a maximum of f. (Identifying maps I \to {\mathbb {R} \choose X} with points x \in X, and maps {\mathbb {R} \choose X} \to I with functions X \to \mathbb {R})
  2. In the same category, for each r \in \mathbb {R}, there is a selection function given by \epsilon (x,f) \Leftrightarrow f(x) \geq r. This corresponds to satisficing at the value r (Reference [simon-environment-satisficing])---that is, selecting any strategy which achieves this value or greater.
  3. For \mathcal {C} = \mathsf {Optic}(\mathcal {C}'), with \mathcal {C}' semiCartesian, there is a selection function \epsilon (x,k) \Leftrightarrow kx = x (using the same identification as above). The agents with this selection function are called predicting agents in Reference [bolt-hedges-winschel-predicting]. These agents attempt to predict the value the environment will return to them.
  4. For \mathcal {C} = \mathsf {Bun}^\mathrm {fop}, the fiberwise opposite of manifolds and smooth bundles---that is, the category of lenses between smooth manifolds---there is a selection function on the tangent bundle TX given by \epsilon (x,k) if and only if k(x) = 0---that is, if x is a fixpoint of the dynamical system identified by k: X \to TX.
  5. Let \mathcal {A} \to \mathcal {C} be a dynamical systems theory and work in the category of lenses. Suppose the monoidal structure on \mathcal {A} is Cartesian, so that a lens I \to {A \choose X} is the same as a map * \to X. Note that I = T(*) has a distinguished section given by the identity. Then there is a selection function on each object TX where a lens I \to TX, given by x: * \to X, is in equilibrium with respect to a lens TX \leftrightarrows I if x is a trajectory between those systems---that is, if it is an equilibrium state of the smooth dynamical system TX \leftrightarrows I. This subsumes the two previous examples.