Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
Equality and Equivalence
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
identity type, equivalence in homotopy type theory
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
natural equivalence, natural isomorphism
principle of equivalence
fiber product, pullback
linear equation, differential equation, ordinary differential equation, critical locus
Euler-Lagrange equation, Einstein equation, wave equation
Schrödinger equation, Knizhnik-Zamolodchikov equation, Maurer-Cartan equation, quantum master equation, Euler-Arnold equation, Fuchsian equation, Fokker-Planck equation, Lax equation
In the category Set a ‘pullback’ is a subset of the cartesian product of two sets. Given a diagram of sets and functions like this:
the ‘pullback’ of this diagram is the subset consisting of pairs such that the equation hold.
A pullback is therefore the categorical semantics of an equation.
This construction comes up, for example, when and are fiber bundles over : then as defined above is the product of and in the category of fiber bundles over . For this reason, a pullback is sometimes called a fibered product (or fiber product or fibre product).
In this case, the fiber of over a (generalized) element of is the ordinary product of the fibers of and over . In other words, the fiber product is the product taken fiber-wise. Of course, the fiber of at the generalized element is itself a fibre product ; the terminology depends on your point of view.
Note that there are maps , sending any to and , respectively. These maps make this square commute:
In fact, the pullback is the universal solution to finding a commutative square like this. In other words, given any commutative square
there is a unique function such that
Since this universal property expresses the concept of pullback purely arrow-theoretically, we can formulate it in any category. It is, in fact, a simple special case of a limit.
In category theory
A pullback is a limit of a diagram like this:
Such a diagram is also called a pullback diagram* or a cospan. If the limit exists, we obtain a commutative square
and the object is also called the pullback. It is well defined up to unique isomorphism. It has the universal property already described above in the special case of the category .
The last commutative square above is called a pullback square.
The concept of pullback is dual to the concept of pushout: that is, a pullback in is the same as a pushout in the opposite category .
In type theory
In type theory a pullback in
is given by the dependent sum over the dependent equality type
As an equalizer
If products exist in , then the pullback
is equivalently the equalizer
of the two morphisms induced by and out of the product of with .
Pasting of pullbacks
Consider a pasting diagram of the form
There are three commuting squares: the two inner ones and the outer one.
Suppose the right-hand inner square is a pullback: then the left-hand one is a pullback if and only if the outer square is.
Proof. Pasting a morphism with the outer square gives rise to a commuting square over the (composite) bottom and right edges of the diagram. The square over the cospan in the left-hand inner square arising from includes a morphism into , which if is a pullback induces the same commuting square over and . So one square is universal iff the other is.
The converse implication does not hold: it may happen that the outer and the left square are pullbacks, but not the right square.
For instance let be a split monomorphism with retract and consider
Then the left square and the outer rectangle are pullbacks but the right square cannot be a pullback unless was already an isomorphism.
The saturation of the class of pullbacks is the class of limits over categories whose groupoid reflection is trivial and such that is L-finite.
- Robert Paré, Simply connected limits. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, CMS