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| In [[category theory]], if C is a [[category (mathematics)|category]] and <math>F: C \to \mathbf{Set}</math> is a set-valued [[functor (category theory)|functor]], the '''category of elements''' of F <math>\mathop{\rm el}(F)</math> (also denoted by ∫<sup>C</sup>F) is the category defined as follows:
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| * Objects are pairs <math>(A,a)</math> where <math>A \in \mathop{\rm Ob}(C)</math> and <math>a \in FA</math>.
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| * An arrow <math>(A,a) \to (B,b)</math> is an arrow <math>f: A \to B</math> in C such that <math>(Ff)a = b</math>.
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| A more concise way to state this is that the category of elements of F is the [[comma category]] <math>\ast\downarrow F</math>, where <math>\ast</math> is a one-point set. The category of elements of F comes with a natural projection <math>\mathop{\rm el}(F) \to C</math> that sends an object (A,a) to A, and an arrow <math>(A,a) \to (B,b)</math> to its underlying arrow in C.
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| == The category of elements of a presheaf ==
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| Somewhat confusingly in some texts (e.g. Mac Lane, Moerdijk), the category of elements for a presheaf is defined differently. If <math>P \in\hat C := \mathbf{Set}^{C^{op}}</math> is a [[presheaf (category theory)|presheaf]], the '''category of elements''' of P (again denoted by <math>\mathop{\rm el}(P)</math>, or, to make the distinction to the above definition clear, ∫<sub>C</sub> P) is the category defined as follows:
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| * Objects are pairs <math>(A,a)</math> where <math>A \in \mathop{\rm Ob}(C)</math> and <math>a\in P(A)</math>.
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| * An arrow <math>(A,a)\to (B,b)</math> is an arrow <math>f:A\to B</math> in C such that <math>(Pf)b = a</math>.
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| As one sees, the direction of the arrows is reversed. One can, once again, state this definition in a more concise manner: the category just defined is nothing but <math>(\ast\downarrow P)^{\rm op}</math>. Consequentially, in the spirit of adding a "co" in front of the name for a construction to denote its opposite, one should rather call this category the category of coelements of P.
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| For C [[small category|small]], this construction can be extended into a functor ∫<sub>C</sub> from <math>\hat C</math> to <math>\mathbf{Cat}</math>, the [[category of small categories]]. In fact, using the [[Yoneda lemma]] one can show that ∫<sub>C</sub>P <math>\cong \mathop{\textbf{y}}\downarrow P</math>, where <math>\mathop{\textbf{y}}: C \to \hat{C}</math> is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫<sub>C</sub> is naturally isomorphic to <math>\mathop{\textbf{y}}\downarrow-: \hat C \to \textbf{Cat}</math>.
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| == References ==
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| *{{cite book|last=Mac Lane|first=Saunders|title=[[Categories for the Working Mathematician]]|publisher=Springer-Verlag|date=1998|edition=2nd|series=Graduate Texts in Mathematics 5|authorlink=Saunders Mac Lane|isbn=0-387-98403-8}}
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| *{{cite book|title=Sheaves in Geometry and Logic|last1=Mac Lane |first1=Saunders |last2=Moerdijk |first2=Ieke |publisher=Springer-Verlag|date=1992|edition=corrected|series=Universitext| isbn=0-387-97710-4|}}
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| ==External links==
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| * {{nlab|id=category+of+elements|title=Category of elements}}
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| [[Category:Representable functors]]
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