# Pullback (category theory)

Last updated

In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms f : X  Z and g : Y  Z with a common codomain. The pullback is often written

## Contents

P = X ×ZY

and comes equipped with two natural morphisms P  X and P  Y. The pullback of two morphisms f and g need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, X ×ZY may intuitively be thought of as consisting of pairs of elements (x, y) with x in X, y in Y, and f(x)  =  g(y). For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square.

The dual concept of the pullback is the pushout .

## Universal property

Explicitly, a pullback of the morphisms f and g consists of an object P and two morphisms p1 : P  X and p2 : P  Y for which the diagram

commutes. Moreover, the pullback (P, p1, p2) must be universal with respect to this diagram. [1] That is, for any other such triple (Q, q1, q2) where q1 : Q  X and q2 : Q  Y are morphisms with fq1 = gq2, there must exist a unique u : Q  P such that

${\displaystyle p_{1}\circ u=q_{1},\qquad p_{2}\circ u=q_{2}.}$

This situation is illustrated in the following commutative diagram.

As with all universal constructions, a pullback, if it exists, is unique up to isomorphism. In fact, given two pullbacks (A, a1, a2) and (B, b1, b2) of the same cospan X  Z  Y, there is a unique isomorphism between A and B respecting the pullback structure.

## Pullback and product

The pullback is similar to the product, but not the same. One may obtain the product by "forgetting" that the morphisms f and g exist, and forgetting that the object Z exists. One is then left with a discrete category containing only the two objects X and Y, and no arrows between them. This discrete category may be used as the index set to construct the ordinary binary product. Thus, the pullback can be thought of as the ordinary (Cartesian) product, but with additional structure. Instead of "forgetting" Z, f, and g, one can also "trivialize" them by specializing Z to be the terminal object (assuming it exists). f and g are then uniquely determined and thus carry no information, and the pullback of this cospan can be seen to be the product of X and Y.

## Examples

### Commutative rings

In the category of commutative rings (with identity), the pullback is called the fibered product. Let A, B, and C be commutative rings (with identity) and α : AC and β : BC (identity preserving) ring homomorphisms. Then the pullback of this diagram exists and given by the subring of the product ring A × B defined by

${\displaystyle A\times _{C}B=\left\{(a,b)\in A\times B\;{\big |}\;\alpha (a)=\beta (b)\right\}}$

along with the morphisms

${\displaystyle \beta '\colon A\times _{C}B\to A,\qquad \alpha '\colon A\times _{C}B\to B}$

given by ${\displaystyle \beta '(a,b)=a}$ and ${\displaystyle \alpha '(a,b)=b}$ for all ${\displaystyle (a,b)\in A\times _{C}B}$. We then have

${\displaystyle \alpha \circ \beta '=\beta \circ \alpha '.}$

### Groups, Modules

In complete analogy to the example of commutative rings above, one can show that all pullbacks exist in the category of groups and in the category of modules over some fixed ring.

### Sets

In the category of sets, the pullback of functions f : X  Z and g : Y  Z always exists and is given by the set

${\displaystyle X\times _{Z}Y=\{(x,y)\in X\times Y|f(x)=g(y)\}=\bigcup _{z\in f(X)\cap g(Y)}f^{-1}[\{z\}]\times g^{-1}[\{z\}],}$

together with the restrictions of the projection maps π1 and π2 to X ×Z Y.

Alternatively one may view the pullback in Set asymmetrically:

${\displaystyle X\times _{Z}Y\cong \coprod _{x\in X}g^{-1}[\{f(x)\}]\cong \coprod _{y\in Y}f^{-1}[\{g(y)\}]}$

where ${\displaystyle \coprod }$ is the disjoint union of sets (the involved sets are not disjoint on their own unless f resp. g is injective). In the first case, the projection π1 extracts the x index while π2 forgets the index, leaving elements of Y.

This example motivates another way of characterizing the pullback: as the equalizer of the morphisms f  p1, g  p2 : X × Y  Z where X × Y is the binary product of X and Y and p1 and p2 are the natural projections. This shows that pullbacks exist in any category with binary products and equalizers. In fact, by the existence theorem for limits, all finite limits exist in a category with binary products and equalizers; equivalently, all finite limits exist in a category with terminal object and pullbacks (by the fact that binary product = pullback on the terminal object, and that an equalizer is a pullback involving binary product).

### Fiber bundles

Another example of a pullback comes from the theory of fiber bundles: given a bundle map π : EB and a continuous map f : X  B, the pullback (formed in the category of topological spaces with continuous maps) X ×B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.

### Preimages and intersections

Preimages of sets under functions can be described as pullbacks as follows:

Suppose f : AB, B0B. Let g be the inclusion map B0B. Then a pullback of f and g (in Set) is given by the preimage f−1[B0] together with the inclusion of the preimage in A

f−1[B0] ↪ A

and the restriction of f to f−1[B0]

f−1[B0] → B0.

Because of this example, in a general category the pullback of a morphism f and a monomorphism g can be thought of as the "preimage" under f of the subobject specified by g. Similarly, pullbacks of two monomorphisms can be thought of as the "intersection" of the two subobjects.

### Least common multiple

Consider the multiplicative monoid of positive integers Z+ as a category with one object. In this category, the pullback of two positive integers m and n is just the pair (LCM(m, n)/m, LCM(m, n)/n), where the numerators are both the least common multiple of m and n. The same pair is also the pushout.

## Properties

• In any category with a terminal object T, the pullback X ×T Y is just the ordinary product X × Y. [2]
• Monomorphisms are stable under pullback: if the arrow f in the diagram is monic, then so is the arrow p2. Similarly, if g is monic, then so is p1. [3]
• Isomorphisms are also stable, and hence, for example, X ×X YY for any map Y  X (where the implied map X  X is the identity).
• In an abelian category all pullbacks exist, [4] and they preserve kernels, in the following sense: if
is a pullback diagram, then the induced morphism ker(p2)  ker(f) is an isomorphism, [5] and so is the induced morphism ker(p1)  ker(g). Every pullback diagram thus gives rise to a commutative diagram of the following form, where all rows and columns are exact:
${\displaystyle {\begin{array}{ccccccc}&&&&0&&0\\&&&&\downarrow &&\downarrow \\&&&&L&=&L\\&&&&\downarrow &&\downarrow \\0&\rightarrow &K&\rightarrow &P&\rightarrow &Y\\&&\parallel &&\downarrow &&\downarrow \\0&\rightarrow &K&\rightarrow &X&\rightarrow &Z\end{array}}}$
Furthermore, in an abelian category, if X  Z is an epimorphism, then so is its pullback P  Y, and symmetrically: if Y  Z is an epimorphism, then so is its pullback P  X. [6] In these situations, the pullback square is also a pushout square. [7]
• There is a natural isomorphism (A×CBBDA×CD. Explicitly, this means:
• if maps f : AC, g : BC and h : DB are given and
• the pullback of f and g is given by r : PA and s : PB, and
• the pullback of s and h is given by t : QP and u : QD ,
• then the pullback of f and gh is given by rt : QA and u : QD.
Graphically this means that two pullback squares, placed side by side and sharing one morphism, form a larger pullback square when ignoring the inner shared morphism.
${\displaystyle {\begin{array}{ccccc}Q&{\xrightarrow {t}}&P&{\xrightarrow {r}}&A\\\downarrow _{u}&&\downarrow _{s}&&\downarrow _{f}\\D&{\xrightarrow {h}}&B&{\xrightarrow {g}}&C\end{array}}}$
• Any category with pullbacks and products has equalizers.

## Weak pullbacks

A weak pullback of a cospan X  Z  Y is a cone over the cospan that is only weakly universal, that is, the mediating morphism u : Q  P above is not required to be unique.

## Notes

1. Mitchell, p. 9
3. Mitchell, p. 9
4. Mitchell, p. 32
5. Mitchell, p. 15
6. Mitchell, p. 34
7. Mitchell, p. 39

## Related Research Articles

In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

In category theory, a branch of mathematics, a universal property is an important property which is satisfied by a universal morphism. Universal morphisms can also be thought of more abstractly as initial or terminal objects of a comma category. Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated.

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.

In category theory, an epimorphism is a morphism f : XY that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: YZ,

In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.

An exact sequence is a sequence of morphisms between objects such that the image of one morphism equals the kernel of the next.

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation.

In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

In mathematics, an equaliser is a set of arguments where two or more functions have equal values. An equaliser is the solution set of an equation. In certain contexts, a difference kernel is the equaliser of exactly two functions.

In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.

In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphismf from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,

In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : ZX and g : ZY with a common domain. The pushout consists of an object P along with two morphisms XP and YP that complete a commutative square with the two given morphisms f and g. In fact, the defining universal property of the pushout essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are and .

In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces.

In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology.

In category theory, a span, roof or correspondence is a generalization of the notion of relation between two objects of a category. When the category has all pullbacks, spans can be considered as morphisms in a category of fractions.

In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.

In algebraic geometry, a derived scheme is a pair consisting of a topological space X and a sheaf of commutative ring spectra on X such that (1) the pair is a scheme and (2) is a quasi-coherent -module. The notion gives a homotopy-theoretic generalization of a scheme.

In algebra, Quillen's Q-construction associates to an exact category an algebraic K-theory. More precisely, given an exact category C, the construction creates a topological space so that is the Grothendieck group of C and, when C is the category of finitely generated projective modules over a ring R, for , is the i-th K-group of R in the classical sense. One puts

In mathematics, specifically in category theory, a quasi-abelian category is a pre-abelian category in which the pushout of a kernel along arbitrary morphisms is again a kernel and, dually, the pullback of a cokernel along arbitrary morphisms is again a cokernel.

## References

• Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). Abstract and Concrete Categories (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN   0-471-60922-6. (now free on-line edition).
• Cohn, Paul M.; Universal Algebra (1981), D. Reidel Publishing, Holland, ISBN   90-277-1213-1 (Originally published in 1965, by Harper & Row).
• Mitchell, Barry (1965). Theory of Categories. Academic Press.