Template:Short description Template:Other Template:Redirect
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 Template:Math and Template:Math with a common codomain. The pullback is written
Usually the morphisms Template:Mvar and Template:Mvar are omitted from the notation, and then the pullback is written
The pullback comes equipped with two natural morphisms Template:Math and Template:Math. The pullback of two morphisms Template:Math and Template:Math need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, Template:Math may intuitively be thought of as consisting of pairs of elements Template:Math with Template:Math in Template:Math, Template:Math in Template:Math, and Template:Math. 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 propertyEdit
Explicitly, a pullback of the morphisms <math>f</math> and <math>g</math> consists of an object <math>P</math> and two morphisms <math>p_1:P\rightarrow X</math> and <math>p_2:P\rightarrow Y</math> for which the diagram
commutes. Moreover, the pullback Template:Math must be universal with respect to this diagram.<ref>Mitchell, p. 9</ref> That is, for any other such triple Template:Math where Template:Math and Template:Math are morphisms with Template:Math, there must exist a unique Template:Math such that
- <math>p_1 \circ u=q_1, \qquad p_2\circ u=q_2.</math>
This situation is illustrated in the following commutative diagram.
.svg){kind=link}
As with all universal constructions, a pullback, if it exists, is unique up to isomorphism. In fact, given two pullbacks Template:Math and Template:Math of the same cospan Template:Math, there is a unique isomorphism between Template:Mvar and Template:Mvar respecting the pullback structure.
Pullback and productEdit
The pullback is similar to the product, but not the same. One may obtain the product by "forgetting" that the morphisms Template:Mvar and Template:Mvar exist, and forgetting that the object Template:Mvar exists. One is then left with a discrete category containing only the two objects Template:Mvar and Template:Mvar, 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" Template:Mvar, Template:Mvar, and Template:Mvar, one can also "trivialize" them by specializing Template:Mvar to be the terminal object (assuming it exists). Template:Mvar and Template:Mvar are then uniquely determined and thus carry no information, and the pullback of this cospan can be seen to be the product of Template:Mvar and Template:Mvar.
ExamplesEdit
Commutative ringsEdit
In the category of commutative rings (with identity), the pullback is called the fibered product. Let Template:Mvar, Template:Mvar, and Template:Mvar be commutative rings (with identity) and Template:Math and Template:Math (identity preserving) ring homomorphisms. Then the pullback of this diagram exists and is given by the subring of the product ring Template:Math defined by
- <math>A \times_{C} B = \left\{(a,b) \in A \times B \; \big| \; \alpha(a) = \beta(b) \right\}</math>
along with the morphisms
- <math>\beta' \colon A \times_{C} B \to A, \qquad \alpha'\colon A \times_{C} B \to B</math>
given by <math>\beta'(a, b) = a</math> and <math>\alpha'(a, b) = b</math> for all <math>(a, b) \in A \times_C B</math>. We then have
- <math>\alpha \circ \beta' = \beta \circ \alpha'.</math>
Groups and modulesEdit
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.
SetsEdit
In the category of sets, the pullback of functions Template:Math and Template:Math always exists and is given by the set
- <math>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\}] ,</math>
together with the restrictions of the projection maps Template:Math and Template:Math to Template:Math.
Alternatively one may view the pullback in Template:Math asymmetrically:
- <math>X\times_Z Y \cong \coprod_{x\in X} g^{-1}[\{f(x)\}] \cong \coprod_{y\in Y} f^{-1}[\{g(y)\}]</math>
where <math>\coprod</math> is the disjoint union of sets (the involved sets are not disjoint on their own unless Template:Mvar resp. Template:Mvar is injective). In the first case, the projection Template:Math extracts the Template:Mvar index while Template:Math forgets the index, leaving elements of Template:Mvar.
This example motivates another way of characterizing the pullback: as the equalizer of the morphisms Template:Math where Template:Math is the binary product of Template:Mvar and Template:Mvar and Template:Math and Template:Math 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 is equal to pullback on the terminal object, and that an equalizer is a pullback involving binary product).
Graphs of functionsEdit
A specific example of a pullback is given by the graph of a function. Suppose that <math>f \colon X \to Y</math> is a function. The graph of Template:Mvar is the set <math display="block">\Gamma_f = \{(x, f(x)) \colon x \in X\} \subseteq X \times Y.</math> The graph can be reformulated as the pullback of Template:Mvar and the identity function on Template:Mvar. By definition, this pullback is <math display="block">X \times_{f,Y,1_Y} Y = \{(x, y) \colon f(x) = 1_Y(y)\} = \{(x, y) \colon f(x) = y\} \subseteq X \times Y,</math> and this equals <math>\Gamma_f</math>.
Fiber bundlesEdit
Another example of a pullback comes from the theory of fiber bundles: given a bundle map Template:Math and a continuous map Template:Math, the pullback (formed in the category of topological spaces with continuous maps) Template:Math is a fiber bundle over Template:Mvar called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles. A special case is the pullback of two fiber bundles Template:Math. In this case Template:Math is a fiber bundle over Template:Math, and pulling back along the diagonal map Template:Math gives a space homeomorphic (diffeomorphic) to Template:Math, which is a fiber bundle over Template:Math. All statements here hold true for differentiable manifolds as well. Differentiable maps Template:Math and Template:Math are transverse if and only if their productTemplate:Math is transverse to the diagonal of Template:Math.<ref>Template:Citation</ref> Thus, the pullback of two transverse differentiable maps into the same differentiable manifold is also a differentiable manifold, and the tangent space of the pullback is the pullback of the tangent spaces along the differential maps.
Preimages and intersectionsEdit
Preimages of sets under functions can be described as pullbacks as follows:
Suppose Template:Math, Template:Math. Let Template:Mvar be the inclusion map Template:Math. Then a pullback of Template:Mvar and Template:Mvar (in Template:Math) is given by the preimage Template:Math together with the inclusion of the preimage in Template:Mvar
and the restriction of Template:Mvar to Template:Math
Because of this example, in a general category the pullback of a morphism Template:Math and a monomorphism Template:Math can be thought of as the "preimage" under Template:Math of the subobject specified by Template:Math. Similarly, pullbacks of two monomorphisms can be thought of as the "intersection" of the two subobjects.
Least common multipleEdit
Consider the multiplicative monoid of positive integers Template:Math as a category with one object. In this category, the pullback of two positive integers Template:Math and Template:Math is just the pair <math>\left(\frac{\operatorname{lcm}(m,n)}{m}, \frac{\operatorname{lcm}(m,n)}{n}\right)</math>, where the numerators are both the least common multiple of Template:Math and Template:Math. The same pair is also the pushout.
PropertiesEdit
- In any category with a terminal object Template:Mvar, the pullback Template:Math is just the ordinary product Template:Math.<ref>Adámek, p. 197.</ref>
- Monomorphisms are stable under pullback: if the arrow Template:Mvar in the diagram is monic, then so is the arrow Template:Math. Similarly, if Template:Mvar is monic, then so is Template:Math.<ref>Mitchell, p. 9</ref>
- Isomorphisms are also stable, and hence, for example, Template:Math for any map Template:Math (where the implied map Template:Math is the identity).
- In an abelian category all pullbacks exist,<ref>Mitchell, p. 32</ref> and they preserve kernels, in the following sense: if
- is a pullback diagram, then the induced morphism Template:Math is an isomorphism,<ref>Mitchell, p. 15</ref> and so is the induced morphism Template:Math. Every pullback diagram thus gives rise to a commutative diagram of the following form, where all rows and columns are exact:
\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}
</math>- Furthermore, in an abelian category, if Template:Math is an epimorphism, then so is its pullback Template:Math, and symmetrically: if Template:Math is an epimorphism, then so is its pullback Template:Math.<ref>Mitchell, p. 34</ref> In these situations, the pullback square is also a pushout square.<ref>Mitchell, p. 39</ref>
- There is a natural isomorphism (A×CB)×B D ≅ A×CD. Explicitly, this means:
- if maps f : A → C, g : B → C and h : D → B are given and
- the pullback of f and g is given by r : P → A and s : P → B, and
- the pullback of s and h is given by t : Q → P and u : Q → D ,
- then the pullback of f and gh is given by rt : Q → A and u : Q → D.
- 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.
\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}
</math>- Any category with pullbacks and products has equalizers.
Weak pullbacksEdit
A weak pullback of a cospan Template:Math is a cone over the cospan that is only weakly universal, that is, the mediating morphism Template:Math above is not required to be unique.
See alsoEdit
NotesEdit
ReferencesEdit
- Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). Abstract and Concrete Categories (4.2MB PDF). Originally publ. John Wiley & Sons. Template:Isbn. (now free on-line edition).
- Cohn, Paul M.; Universal Algebra (1981), D. Reidel Publishing, Holland, Template:Isbn (Originally published in 1965, by Harper & Row).
- Template:Cite book
External linksEdit
- Interactive web page which generates examples of pullbacks in the category of finite sets. Written by Jocelyn Paine.
- Template:Nlab