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In category theory, a branch of mathematics, the category of elements of a presheaf is a category associated to that presheaf whose objects are the elements of sets in the presheaf.
The category of elements of a simplicial set is fundamental in simplicial homotopy theory, a branch of algebraic topology. More generally, the category of elements plays a key role in the proof that every weighted colimit can be expressed as an ordinary colimit, which is in turn necessary for the basic results in theory of pointwise left Kan extensions, and the characterization of the presheaf category as the free cocompletion of a category.
Definition edit
Let be a category and let be a set-valued functor. The category el(F) of elements of F (also denoted ∫C F) is the category whose:
An equivalent definition is that the category of elements of is the comma category ∗↓F, where ∗ is a singleton (a set with one element).
The category of elements of F is naturally equipped with a projection functor Π: ∫C F→C that sends an object (A, a) to A, and an arrow (A,a)→(B,b) to its underlying arrow in C.
As a functor from presheaves to small categories edit
For small C, this construction can be extended into a functor ∫C from Ĉ to Cat, the category of small categories. Using the Yoneda lemma one can show that ∫C P≅y↓P, where y:C→Ĉ is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫C is naturally isomorphic to y↓–:Ĉ→Cat.
See also edit
References edit
- 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.
External links edit
- Category of elements at the nLab