ora-0025
1.13 Stable infinity-categories
An \(\infty \)-category \(\mathcal C\), in the quasi-categorical sense fixed in Definition 1.20, is pointed when it has a zero object: an object \(0\in \mathcal C\) that is both initial and terminal. Equivalently, for every object \(X\), both mapping spaces
are contractible.
A pointed \(\infty \)-category is stable when it has finite limits and finite colimits and a commutative square is a pullback exactly when it is a pushout.
Stability implies that finite limits and finite colimits agree. Consequently, in a presentable stable \(\infty \)-category, filtered colimits commute with finite limits.
Suppose the pointwise right Kan extension uses only finite comma-category shapes. Its values are finite limits. If the online evidence object is the filtered homotopy colimit of its finite stages, stability permits that filtered colimit to pass through the right-Kan limits. This is the key interchange in the derived skeletal-continuity theorem.
Stability is a sufficient hypothesis, not part of the definition of UODL. Decision spaces such as spaces, convex sets, or probability objects need not be stable. Later work should identify weaker target-specific exactness conditions rather than silently assuming stability everywhere.