Maths - Category Theory - Limits and Colimits

Limits and Colimits

A limit is a way to take a diagram and encapsulate its properties into one object as simply as possible.

This is a generalisation that includes:

  • initial/terminal objects
  • (co)products
  • (co)equalizers
  • pullbacks/pushout.
heirchy

Limits

These generate a construction (c) which depends on the diagram such that:

The first condition makes 'c' general enough to capture the essence of the construction.

The second condition makes sure that it is only just general enough (no junk).

Universal Construction Universal Cones over Diagram:

terminal

This is a cone over an empty diagram.

Therefore the diagram puts no constraints on c. It is the most general object and so any other object can map to it.

empty diagram

product

objects 'a' and 'b' are independent of each other so c needs to be general enough to contain the information in both.

More about product on this page.

product diagram limit

arrow

Not a necessarily universal construction but I've included it here for completeness. Here there is an arrow between 'a' and 'b' so they are no longer independent.

 

arrow diagram limit

equaliser

'c' is a subobject of 'a' on which 'f' and 'g' agree. Monics are equalisers in set or any topos.

injective function goes from 'c' to potentially larger objects 'a' and 'b'.

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

 

if h•f=h•g then f=g

(h on left cancels out if monic)

More about equalisers on this page.

pullback

More about pullback on this page.

abc diagram

Colimits

  universal cone over diagram:

Initial

empty diagram

empty diagram

Sum

More about sum on this page.

ab diagram

pushout

abc diagram

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 epi)

Shows which elements of 'a' are equal to other elements of 'a'. (Can be represented by some subset of a×a).

More about co-equalisers on this page.

Limits and CoLimits in Set

Just to give some intuition here are some diagrams showing the internal set structure.

  universal cones over diagram:

terminal

not sure how to show this

 

product

objects 'a' and 'b' are independent of each other so c needs to be general enough to contain the information in both.

arrow injective

In this example we cant make the triangle commute with a 3 element set in cone.

 

 

set arrow injective

arrow surjective

equaliser

'c' is the subset of 'a' on which 'f' and 'g' agree.

co-equaliser

Surjective function merges back those elements that don't commute.

pullback

abc diagram

Relationship to Type Theory

Category theory is deeply related to type theory. We can therefore see these patterns in the constructors and deconstructors(eliminators) of types.

So can we define a type from a diagram? C represents the type we are trying to construct from a diagram.

As an example, look at product types on this page.


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see also:
Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Topoi - Covers more than just topos theory, this is a good introduction to category theory in general.

 

Terminology and Notation

Specific to this page here:

 

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