category of finite Z-sets

 combinatorics  set theory
Notation Z−FinSet\IZ{-}\FinSet Objects finite sets with an action of the additive group Z\IZ, i.e. a permutation Morphisms maps that are compatible with the action, i.e. commute with the permutation Related Z−Set\IZ{-}\Set, M−SetM{-}\Set, FinSet\FinSet, Set\Set

This is a full subcategory of the Grothendieck topos Z−Set\IZ{-}\Set, and provides an example of an elementary topos that has no generator and no cogenerator. Objects can be identified with pairs (X,σ)(X,\sigma), where XX is a finite set and σ:X→X\sigma : X \to X is a permutation of XX. A morphism (X,σ)→(Y,τ)(X,\sigma) \to (Y,\tau) is a map f:X→Yf : X \to Y satisfying τ∘f=f∘σ\tau \circ f = f \circ \sigma.

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: ({∗},id⁡)(\{\ast\},\id)
  • initial object: (∅,id⁡)(\varnothing,\id)
  • products: [finite case] direct product with the obvious action
  • coproducts: [finite case] 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