category of sets with a distinguished subset

Notation

\Mono

Objects pairs

(X, X')

X

is a set and

X' \subseteq X

is a subset Morphisms a morphism

(X, X') \to (Y, Y')

is a function

f : X \to Y

f_*(X') \subseteq Y'

or equivalently

X' \subseteq f^*(Y')

\Set

\Set^{\rightarrow}

External nLab link

This is equivalent to the full subcategory of objects

(X, Y, f)

\Set^{\rightarrow}

f : X \to Y

is an injective function, i.e. a monomorphism. This explains our notation. It is a typical example of a quasitopos.

Satisfied properties

Assigned properties

Deduced properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

isomorphisms: morphisms $f : (X, X') \to (Y, Y')$ such that $f$ is a bijection between $X$ and $Y$ and $f_*(X') = Y'$
monomorphisms: morphisms $f : (X, X') \to (Y, Y')$ such that $f$ is an injective function from $X$ to $Y$
epimorphisms: morphisms $f : (X, X') \to (Y, Y')$ such that $f$ is a surjective function from $X$ to $Y$
regular monomorphisms: morphisms $f : (X, X') \to (Y, Y')$ such that $f$ is an injective function from $X$ to $Y$ and $f^*(Y') = X'$ (in particular, if $f : X \to Y$ is an inclusion map, then this is equivalent to $X' = X \cap Y'$
regular epimorphisms: morphisms $f : (X, X') \to (Y, Y')$ such that $f$ is a surjective function from $X$ to $Y$ and $f_*(X') = Y'$