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category of sets with a distinguished subset

  • notation: Mono\Mono
  • objects: pairs (X,X′)(X, X') where XX is a set and X′⊆XX' \subseteq X is a subset
  • morphisms: a morphism (X,X′)→(Y,Y′)(X, X') \to (Y, Y') is a function f:X→Yf : X \to Y such that f(X′)⊆Y′f(X') \subseteq Y'
  • Related categories: FI\FI, Pos\Pos, Prost\Prost, Set→\Set^{\rightarrow}, Set\Set
  • nLab Link

This is equivalent to the full subcategory of objects (X,Y,f)(X, Y, f) of Set→\Set^{\rightarrow} where f:X→Yf : X \to Y is an injective function, i.e. a monomorphism. This explains our notation.

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: (1,1)(1, 1)
  • initial object: (0,0)(0, 0)
  • products: component-wise defined cartesian product
  • coproducts: component-wise defined disjoint union

Special morphisms

  • isomorphisms: morphisms f:(X,X′)→(Y,Y′)f : (X, X') \to (Y, Y') such that ff is a bijection between XX and YY, and f(X′)=Y′f(X') = Y'
  • monomorphisms: morphisms f:(X,X′)→(Y,Y′)f : (X, X') \to (Y, Y') such that ff is an injective function from XX to YY
  • epimorphisms: morphisms f:(X,X′)→(Y,Y′)f : (X, X') \to (Y, Y') such that ff is a surjective function from XX to YY
  • regular monomorphisms: morphisms f:(X,X′)→(Y,Y′)f : (X, X') \to (Y, Y') such that ff is an injective function from XX to YY, and f−1(Y′)=X′f^{-1}(Y') = X' (in particular, if f:X→Yf : X \to Y is an inclusion map, then this is equivalent to X′=X∩Y′X' = X \cap Y')
  • regular epimorphisms: morphisms f:(X,X′)→(Y,Y′)f : (X, X') \to (Y, Y') such that ff is a surjective function from XX to YY, and f(X′)=Y′f(X') = Y'