category of large families of sets which are mostly empty

 set theory
Notation Set0(I)\Set^{(I)}_0 Objects families of sets X=(Xi)i∈IX = (X_i)_{i \in I} such that supp⁡(X)≔{i∈I:Xi≠∅}\supp(X) \coloneqq \{i \in I : X_i \neq \varnothing\} is essentially small (i.e., isomorphic to a set), where II is a fixed collection that is not essentially small Morphisms families of maps Related Set\Set, Set×Set\Set \times \Set, SetI\Set^I, AbI\Ab^I, VectK(I)\Vect^{(I)}_K, Set1(I)\Set^{(I)}_1

We have added this category solely as an example of a locally cartesian closed category without a generating collection. It is a full subcategory of the product category SetI\Set^I. Most properties are immediately inherited from Set\Set, but there is no terminal object. There are also many differences between this category and its variant Set1(I)\Set^{(I)}_1.

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

—

Special objects

  • initial object: family of empty sets
  • products: [non-empty case] pointwise direct products
  • coproducts: pointwise disjoint unions

Special morphisms

  • isomorphisms: families of bijective maps
  • monomorphisms: families of injective maps
  • epimorphisms: families of surjective maps
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: same as epimorphisms