Maths - Category Theory - Equaliser and Coequaliser

Equaliser

In set an injective function defines a subset relationship. In category theory this can be generalised to a monomorphism (monic).

equaliser  

if h•f=h•g then f=g

(h on left cancels out if it is monic)

Coequaliser

In set an surjective function defines an equivalence relationship. In category theory this can be generalised to a epimorphism (epi).

coequaliser  

if f•h=g•h then f=g

(h on right cancels out if it is epi)

Injective and Surjective Functions in Set Theory

We are not really supposed to look inside sets in category theory but I find it helps me to get some intuition.

Injective Functions

In set an injective function defines a subset relationship. In category theory this can be generalised to a monomorphism (monic).

If the injective function is after the pair then:

h•f=h•g does imply that f=g.

injective

Equaliser

If the injective function is before the pair then:

f•h=g•h does not necessarily imply that f=g.

However an injective function before the pair can form an equaliser, this selects the elements where 'f' and 'g' agree.

Surjective Functions

In set an surjective function defines an equivalence relationship. In category theory this can be generalised to a epimorphism (epi).

If the surjective function is before the pair then:

f•h=g•h does imply that f=g.

Co-equaliser

If the surjective function is after the pair then:

h•f=h•g does not necessarily imply that f=g.

However an surjective function after the pair can form a co-equaliser, merges the elements where 'f' and 'g' do not agree and so the end-to-end function commutes.

Next

This page describes how equalisers can be generalised to limits and coequalisers can be generalised to colimits.


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see also:
  • See this page for introduction to category theory concepts
  • See this page for monic (monomorphism) and epi (epimorphism) functors.
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