Editing Pullback (category theory) (section)

===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>.