category of Z-sets

 combinatorics  set theory
Notation Z−Set\IZ{-}\Set Objects sets with an action of the additive group Z\IZ Morphisms maps that are compatible with the action Parent M−SetM{-}\Set Related Z−FinSet\IZ{-}\FinSet, Set\Set, R−ModR{-}\Mod External nLab Link

This is the special case of the category of MM-sets, where MM is the additive group of integers. Objects can be identified with pairs (X,σ)(X,\sigma), where XX is a set and σ:X→X\sigma : X \to X is a permutation on 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

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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: 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: surjective morphisms

Indistinguishable categories

These categories in the database currently have exactly the same properties as the category of Z-sets. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.