Editing Pullback (category theory)

{{Short description|Most general completion of a commutative square given two morphisms with same codomain}}
{{other|pullback}}
{{Redirect|Fiber product|the case of schemes|Fiber product of schemes}}

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 {{math|''f'' : ''X''&nbsp;→&nbsp;''Z''}} and {{math|''g''&nbsp;:&nbsp;''Y''&nbsp;→&nbsp;''Z''}} with a common codomain. The pullback is written

:{{math|''P'' {{=}} ''X'' ×<sub>''f'', ''Z'', ''g''</sub> ''Y''}}.

Usually the morphisms {{mvar|f}} and {{mvar|g}} are omitted from the notation, and then the pullback is written

:{{math|''P'' {{=}} ''X'' ×<sub>''Z''</sub> ''Y''}}.

The pullback comes equipped with two natural morphisms {{math|''P''&nbsp;→&nbsp;''X''}} and {{math|''P''&nbsp;→&nbsp;''Y''}}.  The pullback of two morphisms {{math|''f''}} and {{math|''g''}} need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, {{math|''X'' ×<sub>''Z''</sub> ''Y''}} may intuitively be thought of as consisting of pairs of elements {{math|(''x'', ''y'')}} with {{math|''x''}} in {{math|''X''}}, {{math|''y''}} in {{math|''Y''}}, and {{math|''f''(''x'')&nbsp; {{=}} &nbsp;''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 diagram|commutative square]].

The [[Dual (category theory)|dual concept]] of the pullback is the ''[[Pushout (category theory)|pushout]]''.

==Universal property==
Explicitly, a pullback of the morphisms <math>f</math> and <math>g</math> consists of an [[Object (category theory)|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

:[[File:Categorical pullback.svg|125px|class=skin-invert]]

[[Commutative diagram|commutes]]. Moreover, the pullback {{math|(''P'', ''p''<sub>1</sub>, ''p''<sub>2</sub>)}} must be [[universal property|universal]] with respect to this diagram.<ref>Mitchell, p. 9</ref> That is, for any other such triple {{math|(''Q'', ''q''<sub>1</sub>, ''q''<sub>2</sub>)}} where {{math|''q''<sub>1</sub> : ''Q''&nbsp;→&nbsp;''X''}} and {{math|''q''<sub>2</sub> : ''Q''&nbsp;→&nbsp;''Y''}} are morphisms with {{math|''f'' ''q''<sub>1</sub>&nbsp;{{=}}&nbsp;''g'' ''q''<sub>2</sub>}}, there must exist a unique {{math|''u''&nbsp;:&nbsp;''Q''&nbsp;→&nbsp;''P''}} 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.
:[[File:Categorical pullback (expanded).svg|225px|class=skin-invert]]

As with all universal constructions, a pullback, if it exists, is unique up to [[isomorphism]].  In fact, given two pullbacks {{math|(''A'', ''a''<sub>1</sub>, ''a''<sub>2</sub>)}} and {{math|(''B'', ''b''<sub>1</sub>, ''b''<sub>2</sub>)}} of the same [[cospan]]  {{math|''X''&nbsp;→&nbsp;''Z''&nbsp;←&nbsp;''Y''}}, there is a unique isomorphism between {{mvar|A}} and {{mvar|B}} respecting the pullback structure.

==Pullback and product==
The pullback is similar to the [[product (category theory)|product]], but not the same.  One may obtain the product by "forgetting" that the morphisms {{mvar|f}} and {{mvar|g}} exist, and forgetting that the object {{mvar|Z}} exists. One is then left with a [[discrete category]] containing only the two objects {{mvar|X}} and {{mvar|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" {{mvar|Z}}, {{mvar|f}}, and {{mvar|g}}, one can also "trivialize" them by specializing {{mvar|Z}} to be the [[terminal object]] (assuming it exists). {{mvar|f}} and {{mvar|g}} are then uniquely determined and thus carry no information, and the pullback of this cospan can be seen to be the product of {{mvar|X}} and {{mvar|Y}}.

==Examples==
===Commutative rings===
[[File:Pullback commutative rings.svg|thumbnail|The category of commutative rings admits pullbacks.]]
In the [[category of commutative rings]] (with identity), the pullback is called the fibered product. Let {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} be [[commutative ring]]s (with identity) and {{math|''α'' : ''A'' → ''C''}} and {{math|''β'' : ''B'' → ''C''}} (identity preserving) [[ring homomorphism]]s. Then the pullback of this diagram exists and is given by the [[subring]] of the [[product ring]] {{math|''A'' × ''B''}} 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 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 {{math|''f'' : ''X''&nbsp;→&nbsp;''Z''}} and {{math|''g''&nbsp;:&nbsp;''Y''&nbsp;→&nbsp;''Z''}} 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 [[Restriction (mathematics)|restrictions]] of the [[projection map]]s {{math|''π''<sub>1</sub>}} and {{math|''π''<sub>2</sub>}} to {{math|''X''&nbsp;×<sub>''Z''</sub>&nbsp;''Y''}}.

Alternatively one may view the pullback in {{math|'''Set'''}} 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 {{mvar|f}} resp. {{mvar|g}} is [[injective]]). In the first case, the projection {{math|''π''<sub>1</sub>}} extracts the {{mvar|x}} index while {{math|''π''<sub>2</sub>}} forgets the index, leaving elements of {{mvar|Y}}.

This example motivates another way of characterizing the pullback: as the [[equaliser (mathematics)|equalizer]] of the morphisms {{math|''f''&nbsp;∘&nbsp;''p''<sub>1</sub>, ''g''&nbsp;∘&nbsp;''p''<sub>2</sub>&nbsp;:&nbsp;''X''&nbsp;×&nbsp;''Y''&nbsp;→&nbsp;''Z''}} where {{math|''X''&nbsp;×&nbsp;''Y''}} is the [[product (category theory)|binary product]] of {{mvar|X}} and {{mvar|Y}} and {{math|''p''<sub>1</sub>}} and {{math|''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 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 functions====
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 {{mvar|f}} 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 {{mvar|f}} and the identity function on {{mvar|Y}}.  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 bundles===
Another example of a pullback comes from the theory of [[fiber bundle]]s: given a bundle map {{math|''π'' : ''E'' → ''B''}} and a [[continuous map]] {{math|''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''B''}}, the pullback (formed in the [[category of topological spaces]] with [[Continuous function (topology)|continuous maps]]) {{math|''X''&nbsp;×<sub>''B''</sub>&nbsp;''E''}} is a fiber bundle over {{mvar|X}} called the [[pullback bundle]]. The associated commutative diagram is a morphism of fiber bundles. A special case is the pullback of two fiber bundles {{math|''E''<sub>''1''</sub>, ''E''<sub>2</sub> → ''B''}}. In this case {{math|''E''<sub>1</sub> × ''E''<sub>2</sub>}} is a fiber bundle over {{math|''B × B''}}, and pulling back along the diagonal map {{math|''B'' → ''B × B''}} gives a space homeomorphic (diffeomorphic) to {{math|''E''<sub>1</sub> ×<sub>''B''</sub> ''E''<sub>2</sub>}}, which is a fiber bundle over {{math|''B''}}. All statements here hold true for differentiable [[Differentiable manifold|manifolds]] as well. Differentiable maps {{math|''f''&nbsp;:&nbsp;''M''&nbsp;→&nbsp;''N''}} and {{math|''g''&nbsp;:&nbsp;''P''&nbsp;→&nbsp;''N''}} are [[Transversality (mathematics)|transverse]] if and only if their product{{math| ''M × P'' → ''N × N''}} is transverse to the diagonal of {{math|''N''}}.<ref>{{Citation |last=Lee |first=John M. |title=Smooth Manifolds |date=2003 |work=Graduate Texts in Mathematics |pages=1–29 |url=https://doi.org/10.1007/978-0-387-21752-9_1 |access-date=2025-02-28 |place=New York, NY |publisher=Springer New York |isbn=978-0-387-95448-6}}</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 intersections===
[[Preimage]]s of sets under functions can be described as pullbacks as follows: 

Suppose {{math|''f'' : ''A'' → ''B''}}, {{math|''B''<sub>0</sub> ⊆ ''B''}}. Let {{mvar|g}} be the [[inclusion map]] {{math|''B''<sub>0</sub> ↪ ''B''}}. Then a pullback of {{mvar|f}} and {{mvar|g}} (in {{math|'''Set'''}}) is given by the preimage {{math|''f''<sup>−1</sup>[''B''<sub>0</sub>]}} together with the inclusion of the preimage in {{mvar|A}}

:{{math|''f''<sup>−1</sup>[''B''<sub>0</sub>] ↪ ''A''}}

and the restriction of {{mvar|f}} to {{math|''f''<sup>−1</sup>[''B''<sub>0</sub>]}}

:{{math|''f''<sup>−1</sup>[''B''<sub>0</sub>] → ''B''<sub>0</sub>}}.

Because of this example, in a general category the pullback of a morphism {{math|''f''}} and a [[monomorphism]] {{math|''g''}} can be thought of as the "preimage" under {{math|''f''}} of the [[subobject]] specified by {{math|''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 [[integer]]s {{math|'''Z'''<sub>+</sub>}} as a category with one object. In this category, the pullback of two positive integers {{math|''m''}} and {{math|''n''}} 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 {{math|''m''}} and {{math|''n''}}. The same pair is also the pushout.

==Properties==
*In any category with a [[terminal object]] {{mvar|T}}, the pullback {{math|''X''&nbsp;×<sub>''T''</sub>&nbsp;''Y''}} is just the ordinary [[product (category theory)|product]] {{math|''X''&nbsp;×&nbsp;''Y''}}.<ref>Adámek, p. 197.</ref>
*[[Monomorphism]]s are stable under pullback: if the arrow {{mvar|f}} in the diagram is monic, then so is the arrow {{math|''p''<sub>2</sub>}}. Similarly,  if {{mvar|g}} is monic, then so is {{math|''p''<sub>1</sub>}}.<ref>Mitchell, p. 9</ref>
*[[Isomorphism]]s are also stable, and hence, for example, {{math|''X''&nbsp;×<sub>''X''</sub>&nbsp;''Y'' ≅ ''Y''}} for any map {{math|''Y''&nbsp;→&nbsp;''X''}} (where the implied map {{math|''X''&nbsp;→&nbsp;''X''}} is the identity).
* In an [[abelian category]] all pullbacks exist,<ref>Mitchell, p. 32</ref> and they preserve [[kernel (category theory)|kernels]], in the following sense: if
::[[File:Categorical pullback.svg|125px|class=skin-invert]]
::
:is a pullback diagram, then the induced morphism {{math|ker(''p''<sub>2</sub>)&nbsp;→&nbsp;ker(''f'')}} is an isomorphism,<ref>Mitchell, p. 15</ref> and so is the induced morphism {{math|ker(''p''<sub>1</sub>)&nbsp;→&nbsp;ker(''g'')}}. Every pullback diagram thus gives rise to a commutative diagram of the following form, where all rows and columns are [[exact sequence|exact]]:
<div class="center"><math>
\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></div>
:Furthermore, in an abelian category, if {{math|''X''&nbsp;→&nbsp;''Z''}} is an epimorphism, then so is its pullback {{math|''P''&nbsp;→&nbsp;''Y''}}, and symmetrically: if {{math|''Y''&nbsp;→&nbsp;''Z''}} is an epimorphism, then so is its pullback {{math|''P''&nbsp;→&nbsp;''X''}}.<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''×<sub>''C''</sub>''B'')×<sub>''B''</sub> ''D'' &cong; ''A''×<sub>''C''</sub>''D''. Explicitly, this means: 
** if maps ''f'' : ''A'' &rarr; ''C'', ''g'' : ''B'' &rarr; ''C'' and ''h'' : ''D'' &rarr; ''B'' are given and 
** the pullback of ''f'' and ''g'' is given by ''r'' : ''P'' &rarr; ''A'' and ''s'' : ''P'' &rarr; ''B'', and
** the pullback of ''s'' and ''h'' is given by  ''t'' : ''Q'' &rarr; ''P'' and ''u'' : ''Q'' &rarr; ''D'' , 
** then the pullback of ''f'' and ''gh'' is given by ''rt'' : ''Q'' &rarr; ''A'' and  ''u'' : ''Q'' &rarr; ''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.
<div class="center"><math>
\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></div>

* Any category with pullbacks and products has equalizers.

== Weak pullbacks ==
A '''weak pullback''' of a [[span (category theory)|cospan]] {{math|''X''&nbsp;→&nbsp;''Z''&nbsp;←&nbsp;''Y''}} is a [[cone (category theory)|cone]] over the cospan that is only [[weakly universal property|weakly universal]], that is, the mediating morphism {{math|''u''&nbsp;:&nbsp;''Q''&nbsp;→&nbsp;''P''}} above is not required to be unique.

==See also==
*[[Pullback (differential geometry)|Pullbacks in differential geometry]]
*[[Relational algebra#θ-join and equijoin|Equijoin]] in [[relational algebra]]
*[[Fiber product of schemes]]

==Notes==
{{reflist}}

==References==
*Adámek, Jiří, [[Horst Herrlich|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).
*[[Paul Cohn|Cohn, Paul M.]]; ''Universal Algebra'' (1981), D. Reidel Publishing, Holland, {{isbn|90-277-1213-1}} ''(Originally published in 1965, by Harper & Row)''.
*{{Cite book|last=Mitchell|first=Barry|title=Theory of Categories|publisher=Academic Press|year=1965}}

==External links==
*[https://web.archive.org/web/20080916162345/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.
* {{Nlab|id=pullback}}

{{Category theory}}

[[Category:Limits (category theory)]]