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Category of elements

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In category theory, if C is a category and is a set-valued functor, the category of elements of F (also denoted by ∫CF) is the category defined as follows:

  • Objects are pairs where and .
  • An arrow is an arrow in C such that .

A more concise way to state this is that the category of elements of F is the comma category , where is a one-point set. The category of elements of F comes with a natural projection that sends an object (A,a) to A, and an arrow to its underlying arrow in C.

The category of elements of a presheaf

In some texts (e.g. Mac Lane, Moerdijk) the category of elements is used for presheaves. We state it explicitly for completeness. If is a presheaf, the category of elements of P (again denoted by , or, to make the distinction to the above definition clear, ∫C P=∫Cop P) is the category defined as follows:

  • Objects are pairs where and .
  • An arrow is an arrow in C such that .

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 . 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.

For C small, this construction can be extended into a functor ∫C from to , the category of small categories. In fact, using the Yoneda lemma one can show that ∫CP , where is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫C is naturally isomorphic to .

The category of elements of an operad algebra

Given a (colored) operad and a functor, also called an algebra, , one obtains a new operad, called the category of elements and denoted , generalizing the above story for categories. It has the following description:

  • Objects are pairs where and .
  • An arrow is an arrow in such that


See also

References

  • Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8.
  • Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in Geometry and Logic. Universitext (corrected ed.). Springer-Verlag. ISBN 0-387-97710-4.