Image (category theory)

In category theory , an image of a morphism is a sub-object of that has the following universal property : ${\displaystyle f\colon X\to Y}$ ${\displaystyle h\colon I\to Y}$ ${\displaystyle Y}$

• There is a morphism with . ${\displaystyle g\colon X\to I}$ ${\displaystyle f=hg}$
• For every sub-object that fulfills the above property ( ), there is a unique morphism with and . ${\displaystyle l\colon Z\to Y}$ ${\displaystyle f=lk}$ ${\displaystyle m\colon I\to Z}$ ${\displaystyle k=mg}$ ${\displaystyle h=lm}$

The cobild of a morphism is the dual term: a cobild is a quotient object of X that has the following universal property: ${\displaystyle f\colon X\to Y}$ ${\displaystyle g\colon X\to C}$

• There is a morphism mi . ${\displaystyle h\colon C\to Y}$ ${\displaystyle f=hg}$
• For every quotient object that fulfills the above property ( ), there is a unique morphism with and . ${\displaystyle k\colon X\to Z}$ ${\displaystyle f=lk}$ ${\displaystyle m\colon Z\to C}$ ${\displaystyle g=mk}$ ${\displaystyle l=hm}$

In categories with a core and a coke core , each core of a coking core of f is an image of f , and each coking core of the core is a cobild.

In Abelian categories such as the categories of vector spaces or Abelian groups, the image and the image match. In the categories mentioned, they are also the same as the set-theoretical picture.