category of F(I)-sets

Unknown properties

There is 1 property for which the database doesn't have an answer if it is satisfied or not. Please help to contribute the data!

• is cototal

Special objects

• terminal object: singleton set with the unique action
• initial object: empty set with the unique action
• products: direct product with the obvious action
• coproducts: disjoint union with the obvious action

Special morphisms

• isomorphisms: bijective morphisms
• monomorphisms: injective morphisms
• epimorphisms: surjective morphisms
• regular monomorphisms: same as monomorphisms
• regular epimorphisms: same as epimorphisms

Comments

• The category has appeared in MO/258269.
• The category is well-defined in our Foundations. For instance, the collection of objects is $\prod_{X \in \SetColl} \prod_{i \in I} \Aut(X)$, which is indeed a collection since collections are closed under the usual operations of set theory. Another explanation is that it is the functor category $[B(F(I)),\Set]$, where $B(F(I))$ is the one-object category associated with $F(I)$.
• Surprisingly, the forgetful functor to $\Set$ does not preserve large limits, hence is not representable, and therefore has no left adjoint. For $i \in I$ consider the object $X_i = (\{0,1\},\sigma^{[i]})$, where $\sigma^{[i]}_i = (0 ~ 1)$ and $\sigma^{[i]}_j = \id$ for $j \neq i$. One can show that the product of the large family $(X_i)_{i \in I}$ exists in $F(I){-}\Set$ and is given by the empty set with the unique action. But the large family of underlying sets has no product in $\Set$.