Ultraproduct
Ultraproduct monad

The ultrafilter monad is the codensity monad of the inclusion of the category of finite sets into the category of all sets.

Similarly, the ultraproduct monad is the codensity monad of the inclusion of the category F i n F a m {\displaystyle \mathbf {FinFam} } of finitely-indexed families of sets into the category F a m {\displaystyle \mathbf {Fam} } of all indexed families of sets. So in this sense, ultraproducts are categorically inevitable. Explicitly, an object of F a m {\displaystyle \mathbf {Fam} } consists of a non-empty index set I {\displaystyle I} and an indexed family (M i) i ∈ I {\displaystyle \left(M_{i}\right)_{i\in I}} of sets. A morphism (N i) j ∈ J → (M i) i ∈ I {\displaystyle \left(N_{i}\right)_{j\in J}\to \left(M_{i}\right)_{i\in I}} between two objects consists of a function ϕ: I → J {\displaystyle \phi:I\to J} between the index sets and a J {\displaystyle J} -indexed family (ϕ j) j ∈ J {\displaystyle \left(\phi _{j}\right)_{j\in J}} of function ϕ j: M ϕ (j) → N j. {\displaystyle \phi _{j}:M_{\phi (j)}\to N_{j}.} The category F i n F a m {\displaystyle \mathbf {FinFam} } is a full subcategory of this category of F a m {\displaystyle \mathbf {Fam} } consisting of all objects (M i) i ∈ I {\displaystyle \left(M_{i}\right)_{i\in I}} whose index set I {\displaystyle I} is finite. The codensity monad of the inclusion map F i n F a m ↪ F a m {\displaystyle \mathbf {FinFam} \hookrightarrow \mathbf {Fam} } is then, in essence, given by (M i) i ∈ I ↦ (∏ i ∈ I M i / U) U ∈ U (I). {\displaystyle \left(M_{i}\right)_{i\in I}~\mapsto ~\left(\prod _{i\in I}M_{i}\,/\,{\mathcal {U}}\right)_{{\mathcal {U}}\in U(I)}\,.}
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