category of connected sequences of sets

 set theory
Notation [(N,≤),Set]conn[(\IN,\leq),\Set]_{\conn} Objects sequences of sets X0→X1→X2→⋯X_0 \to X_1 \to X_2 \to \cdots that are connected, meaning that colim⁡n≥0Xn\colim_{n \geq 0} X_n is a singleton set Morphisms commutative diagrams Related [(N,≤),Set][(\IN,\leq),\Set], Set≠∅\Setne

This is a full subcategory of the category of sequences of sets, consisting of those sequences X0→X1→X2→⋯X_0 \to X_1 \to X_2 \to \cdots whose colimit is a singleton set. Equivalently, the following two conditions are satisfied: (1) There is some n≥0n \geq 0 such that Xn≠∅X_n \neq \varnothing. (2) For all n≥0n \geq 0 and all x,y∈Xnx,y \in X_n, there is some m≥nm \geq n such that xx and yy map to the same element of XmX_m.
We have added this category as an example of a category with a subobject classifier but without an initial object.

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 {∗}→{∗}→⋯\{\ast\} \to \{\ast\} \to \cdots
  • products: [finite case] component-wise defined direct product

Special morphisms

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

Comments

  • This category was suggested by Jonas Frey at MO/515206.