category of sequences of sets

 set theory
Notation [(N,≤),Set][(\IN,\leq),\Set] Objects sequences of sets X0→X1→X2→⋯X_0 \to X_1 \to X_2 \to \cdots Morphisms commutative diagrams Related Set\Set, Set→\Set^{\rightarrow}, [(N,≤),Ab][(\IN,\leq),\Ab], Set×Set\Set \times \Set, [(N,≤),Set]conn[(\IN,\leq),\Set]_{\conn}

As a functor category of Set\Set, most properties are inherited from Set\Set, but a notable difference is that it has no generator.

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: the constant sequence 1→1→⋯1 \to 1 \to \cdots
  • initial object: the constant sequence 0→0→⋯0 \to 0 \to \cdots
  • products: component-wise defined direct product
  • coproducts: component-wise defined disjoint union

Special morphisms

  • isomorphisms: morphisms f=(fn)n≥0f = (f_n)_{n \geq 0} where every fnf_n is bijective
  • monomorphisms: morphisms f=(fn)n≥0f = (f_n)_{n \geq 0} where every fnf_n is injective
  • epimorphisms: morphisms f=(fn)n≥0f = (f_n)_{n \geq 0} where every fnf_n is surjective
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: same as epimorphisms

Indistinguishable categories

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