Definition 6.135.
An ultracategory is a category equipped with coherent ultraproduct operations: for every ultrafilter \(\Uu\) on a set \(I\), there is a functor
\[P^{\Uu}\colon C^I \to C.\]
More precisely, let \(\mathrm{Stone}^{\mathrm{free}}\) denote the category of Stone–Čech compactifications of sets. The category \(\Cat^{\ultra}\) of ultracategories is the subcategory of \(\mathrm{LaxFun}((\mathrm{Stone}^{\mathrm{free}})\catop,\Cat)\) spanned by the lax functors \(F\) satisfying:
We have \(F(\beta I) \simeq \prod_I F(*)\).
Given \(f\colon \beta I \to \beta J\) and \(g\colon \beta J \to \beta K\), if \(f\) is induced by a map \(I \to J\), then \(f^* \circ g^* \simeq (g \circ f)^*\).