category of large families of sets which are mostly singletons

 set theory
Notation Set1(I)\Set^{(I)}_1 Objects families of sets X=(Xi)i∈IX = (X_i)_{i \in I} such that S(X)≔{i∈I:Xi≇1}S(X) \coloneqq \{i \in I : X_i \not\cong 1\} 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, Set0(I)\Set^{(I)}_0, SetI\Set^I, AbI\Ab^I, VectK(I)\Vect^{(I)}_K
We have added this category solely as an example of a cartesian closed category without a generating collection, but it also satisfies some other interesting combinations of properties. It is a full subcategory of the product category SetI\Set^I. Most properties are immediately inherited from Set\Set, but there is no initial object. There are also many differences between this category and its variant Set0(I)\Set^{(I)}_0.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: family of singleton sets
  • products: pointwise direct products

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