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| In [[category theory]], a branch of [[mathematics]], a '''pullback''' (also called a '''fiber product''', '''fibre product''', '''fibered product''' or '''Cartesian square''') is the [[limit (category theory)|limit]] of a [[diagram (category theory)|diagram]] consisting of two [[morphism]]s ''f'' : ''X'' → ''Z'' and ''g'' : ''Y'' → ''Z'' with a common codomain; it is the limit of the [[cospan]] <math>X \rightarrow Z \leftarrow Y</math>. The pullback is often written
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| :<math> P = X \times_Z Y.\, </math>
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| ==Universal property==
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| Explicitly, the pullback of the morphisms ''f'' and ''g'' consists of an object ''P'' and two morphisms ''p''<sub>1</sub> : ''P'' → ''X'' and ''p''<sub>2</sub> : ''P'' → ''Y'' for which the diagram
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| :[[Image:Categorical pullback.svg|125px]]
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| [[commutative diagram|commutes]]. Moreover, the pullback (''P'', ''p''<sub>1</sub>, ''p''<sub>2</sub>) must be [[universal property|universal]] with respect to this diagram. That is, for any other such triple (''Q'', ''q''<sub>1</sub>, ''q''<sub>2</sub>) for which the following diagram commutes, there must exist a unique ''u'' : ''Q'' → ''P'' (called a mediating morphism) such that <math>p_2 \circ u=q_2</math> and <math>p_1\circ u=q_1</math>
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| :[[Image:Categorical pullback (expanded).svg|225px]]
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| As with all universal constructions, the pullback, if it exists, is unique up to [[isomorphism]]. In fact, given two pullbacks (''A'', ''a''<sub>1</sub>, ''a''<sub>2</sub>) and (''B'', ''b''<sub>1</sub>, ''b''<sub>2</sub>) of the same cospan, there is a unique isomorphism between ''A'' and ''B'' respecting the pullback structure.
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| ==Weak pullbacks== | |
| A '''weak pullback''' of a [[span (category theory)|cospan]] ''X'' → ''Z'' ← ''Y'' is a [[cone (category theory)|cone]] over the cospan that is only [[weakly universal property|weakly universal]], that is, the mediating morphism ''u'' : ''Q'' → ''P'' above is not required to be unique.
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| ==Discussion==
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| The pullback is similar to the [[product (category theory)|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''.
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| The [[dual (category theory)|categorical dual]] of a pullback is a called a ''[[pushout (category theory)|pushout]]''. Remarks opposite to the above apply: the pushout is a [[coproduct]] with additional structure.
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| ==Examples==
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| 1. In the category of commutative rings (with identity), denoted '''CRing''', the pullback is called the fibered product. Let <math> A, B, C \in Ob(\mathbf{CRing}) </math> and <math>\alpha\colon A \to C, \beta\colon B \to C \in \mathsf{Hom}(\mathbf{CRing}) </math>, that is, A, B, and C are commutative rings with identity and <math>\alpha</math> and <math>\beta</math> are ring homomorphisms. Then the pullback of these objects and morphisms is defined to be the subset of the Cartesian product <math> A \times B </math> defined by
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| :<math> A \times_{C} B = \left\{(a,b) \in A \times B \; \big| \; \alpha(a) = \beta(b) \right\} </math>
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| along with the morphisms <math> \beta' \colon A \times_{C} B \to A </math> and <math>\alpha'\colon A \times_{C} B \to B </math> such that <math>\alpha \circ \beta' = \beta \circ \alpha' </math>.
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| 2. In the [[category of sets]], a pullback of ''f'' and ''g'' is given by the set
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| :<math>X\times_Z Y = \{(x, y) \in X \times Y| f(x) = g(y)\},\,</math>
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| together with the restrictions of the [[projection map]]s <math>\pi_1</math> and <math>\pi_2</math> to ''X'' ×<sub>''Z''</sub> ''Y'' .
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| Alternatively one may view the pullback in '''Set''' asymmetrically:
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| :<math>X\times_Z Y</math>
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| :<math>
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| \cong
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| \coprod_{x\in X} g^{-1}[\{f(x)\}]</math>
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| :<math>
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| \cong
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| \coprod_{y\in Y} f^{-1}[\{g(y)\}]
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| </math>
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| where <math>\coprod</math> is the ''disjoint'' (tagged) union of sets (the involved sets are not disjoint on their own unless ''f'' resp. ''g'' is injective). In the first case, the projection <math>\pi_1</math> extracts the ''x'' index while <math>\pi_2</math> forgets the index, leaving elements of ''Y''.
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| *This example motivates another way of characterizing the pullback: as the [[equaliser (mathematics)|equalizer]] of the morphisms ''f'' <small>o</small> ''p''<sub>1</sub>, ''g'' <small>o</small> ''p''<sub>2</sub> : ''X'' × ''Y'' → ''Z'' where ''X'' × ''Y'' is the [[product (category theory)|binary product]] of ''X'' and ''Y'' and ''p''<sub>1</sub> and ''p''<sub>2</sub> 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 a terminal object, binary products and equalizers.
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| 3. Another example of a pullback comes from the theory of [[fiber bundle]]s: given a bundle map π : ''E'' → ''B'' and a [[continuous map]] ''f'' : ''X'' → ''B'', the pullback ''X'' ×<sub>''B''</sub> ''E'' is a fiber bundle over ''X'' called the [[pullback bundle]]. The associated commutative diagram is a morphism of fiber bundles.
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| 4. In any category with a [[terminal object]] ''Z'', the pullback ''X'' ×<sub>''Z''</sub> ''Y'' is just the ordinary [[product (category theory)|product]] ''X'' × ''Y''.<ref>Adámek, p. 197.</ref>
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| === Preimages ===
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| [[Preimage]]s of sets under functions can be described as pullbacks as follows: Suppose
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| : ''f'' : ''A'' → ''B''
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| and
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| : ''B<sub>0</sub>'' ⊆ B.
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| Let ''g'' be the inclusion map ''B<sub>0</sub>'' ↪ ''B''.
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| Then a pullback of ''f'' and ''g'' (in '''Set''') is given by the preimage ''f''<sup>-1</sup> [ ''B<sub>0</sub>'' ] together with the inclusion of the preimage in ''A''
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| : ''f''<sup>-1</sup> [ ''B<sub>0</sub>'' ] ↪ ''A''
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| and the restriction of ''f'' to ''f''<sup>-1</sup> [ ''B<sub>0</sub>'' ] | |
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| : ''f''<sup>-1</sup> [ B<sub>0</sub> ]'' → ''B<sub>0</sub>''.
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| ==Properties==
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| *Whenever ''X'' ×<sub>''Z''</sub> ''Y'' exists, then so does ''Y'' ×<sub>''Z''</sub> ''X'' and there is an isomorphism ''X'' ×<sub>''Z''</sub> ''Y'' <math> \cong</math> ''Y'' ×<sub>''Z''</sub> ''X''.
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| *[[Monomorphism]]s are stable under pullback: if the arrow ''f'' above is monic, then so is the arrow ''p''<sub>2</sub>. For example, in the category of sets, if ''X'' is a subset of ''Z'', then, for any ''g'' : ''Y'' → ''Z'', the pullback ''X'' ×<sub>''Z''</sub> ''Y'' is the [[inverse image]] of ''X'' under ''g''.
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| *[[Isomorphism]]s are also stable, and hence, for example, ''X'' ×<sub>''X''</sub> ''Y'' <math>\cong</math> ''Y'' for any map ''Y'' → ''X''.
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| * Any category with pullbacks and products has equalizers.
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| ==See also==
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| *[[Pullback (differential geometry)|Pullbacks in differential geometry]]
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| *[[Relational algebra#θ-join and equijoin|Equijoin]] in [[relational algebra]].
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). [http://katmat.math.uni-bremen.de/acc/acc.pdf ''Abstract and Concrete Categories''] (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).
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| *Cohn, Paul M.; ''Universal Algebra'' (1981), D.Reidel Publishing, Holland, ISBN 90-277-1213-1 ''(Originally published in 1965, by Harper & Row)''.
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| ==External links==
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| *[http://www.j-paine.org/cgi-bin/webcats/webcats.php Interactive Web page ] which generates examples of pullbacks in the category of finite sets. Written by Jocelyn Paine.
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| *[http://ncatlab.org/nlab/show/pullback pullbacks ] on the N-Category Lab.
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| [[Category:Limits (category theory)]]
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