Editing Pullback (category theory) (section)

==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.