|Given fixed objects A,B,C and morphisms g,f then a pullback is P with some universal property.|
Pullback is a:
- Generalisation of both intersection an inverse.
- If a category has binary products and equalisers, then it has pullbacks.
In programming terms it is related to the concept of indexing.
The pullback is like a type of division for function composition, in other words if multiplication is function composition then what is division? This comes in left and right flavors that is:
if h = g o f
- We know f and h, how do we find g?
- We know g and h, how do we find f?
|universal cone over diagram|
P is the pullback of f along g or g along f (A,B,C fixed) .
Square must commute (in best possible way) so any other square, also containing (A,B,C) must uniquely map to it.
P is the pushout of A,B,C
||a kind of colimit|
k(x) maps to <j(x),i(x)>
called a fibred product.
|A U B /equivalence relation|
In concrete categories the Cartesian product is often the categorical product. The pullback is like the categorical product but with additional conditions. So in the following examples we have a Cartesian product with a subset of the rows and columns.
Examples in Set
Set Example 1
|This diagram is both a pullback and a pushout.|
Set Example 2
How does the above example relate to the idea that the pullback is a subset of the cartesian product of two sets? We have this restriction on the Cartesian product:
|So, in this example, 'P' contains only the subset of the Cartesian product shown in red (that is <e,e>).|
Example in Number Expressions
This generates pairs of elements <a, b> : P with a in A, b in B, such that f(a) = g(b).
So the requirement for this square to commute generates in P all the possible pairs of numbers that are equal to each other.
So, in the example on the right, 4 is a 'representative' for all the things it is equal to such as 2+2 and 3+1. Sometimes the notation  is used for the set of all the things it is equal to (although this notation could be confused with list).
More about this on Leibniz equality page.
Examples in Graphs
- Pullbacks always exist.
- Pushouts do not necessarily exist.
In this example only the red arrows go both ways round the square so only the red arrows exist in 'G'.
Examples in Groups
Example - Type Theory - Substitution as Pullback
- Substitution and adjunction discussed on this page.
- Pullback and equality discussed on this page.
- Equality discussed on this page.
Substitution as Pullback
Can we express substitution in category theory terms? See discussion on page here.
|Substitution of a term into a predicate is pullback, but substitution of a term into a term is composition.|
|Type theory version||Substitution of a term into a dependant type is pullback, but substitution of a term into a term is composition.|
There is a discussion of this topic on this site.
So why the difference? I think in the predicate case we are going backwards.
Typically a predicate could be an equation which may be true or false depending on the values of its variables.
Or a function from a variable to bool (or some truth object).
Pullback as generalisation of Inverse
Hence we have two morphisms to C so this looks like the given part of a pullback.
|To complete the square we add cone for this diagram and we call it f-1(B) to emphasise that it is the inverse of 'f'.|
Reversing a function can generate interesting structure, for example, reversing addition on natural numbers leads to the integers and reversing multipication leads to fractional numbers.
|If we reverse a surjective mapping, where many elements in the domain map to a single element in the codomain, then reversing the function gives a fibre bundle.|
|If we reverse a injective mapping, where not all elements in the codomain are mapped to, then reversing the function gives a subobject structure.|
Relationship to Equaliser
If the category A is the same as the category B then the pullback becomes an equaliser.
See equalisers on this page.
Examples in Various Categories
|generalisation||a kind of limit|
|group||the product is given by the cartesian product with multiplication defined componentwise.|
|Grp (abelian)||direct sum|
|vector space||direct sum|
|poset||greatest lower bound
|base topological space|
greatest lower bounds (meets)
|Top||the space whose underlying set is the cartesian product and which carries the product topology|
tensor products are not categorial products.
In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point).