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Grothendieck construction

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The Grothendieck construction (named after Alexander Grothendieck) is a construction used in the mathematical field of category theory. It is a fundamental construction in the theory of descent, in the theory of stacks, and in fibred category theory. In categorical logic, the construction is used to model the relationship between a type theory and a logic over that type theory, and allows for the translation of concepts from indexed category theory into fibred category theory, such as Lawvere's concept of hyperdoctrine.

The Grothendieck construction was first studied for the special case presheaves of sets by Mac Lane, where it was called the category of elements.[1]

Motivation

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If is a family of sets indexed by another set, one can form the disjoint union or coproduct

,

which is the set of all ordered pairs such that . The disjoint union set is naturally equipped with a "projection" map

defined by

.

From the projection it is possible to reconstruct the original family of sets up to a canonical bijection, as for each via the bijection . In this context, for , the preimage of the singleton set is called the "fiber" of over , and any set equipped with a choice of function is said to be "fibered" over . In this way, the disjoint union construction provides a way of viewing any family of sets indexed by as a set "fibered" over , and conversely, for any set fibered over , we can view it as the disjoint union of the fibers of . Jacobs has referred to these two perspectives as "display indexing" and "pointwise indexing".[2]

The Grothendieck construction generalizes this to categories. For each category , family of categories indexed by the objects of in a functorial way, the Grothendieck construction returns a new category fibered over by a functor whose fibers are the categories .

Definition

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Let be a functor from any small category to the category of small categories. The Grothendieck construction for is the category (also written , or ), with

  • objects being pairs , where and ; and
  • morphisms in being pairs such that in , and in .

Composition of morphisms is defined by .

Example

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If is a group, then it can be viewed as a category, with one object and all morphisms invertible. Let be a functor whose value at the sole object of is the category a category representing the group in the same way. The requirement that be a functor is then equivalent to specifying a group homomorphism where denotes the group of automorphisms of Finally, the Grothendieck construction, results in a category with one object, which can again be viewed as a group, and in this case, the resulting group is (isomorphic to) the semidirect product

See also

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References

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  • Mac Lane and Moerdijk, Sheaves in Geometry and Logic, pp. 44.
  • R. W. Thomason (1979). Homotopy colimits in the category of small categories. Mathematical Proceedings of the Cambridge Philosophical Society, 85, pp 91–109. doi:10.1017/S0305004100055535.
Specific
  1. ^ Mac Lane, Saunders; Moerdijk, Ieke (1994). Sheaves in geometry and logic: a first introduction to topos theory (2., corr. print ed.). New York: Springer. ISBN 9780387977102.
  2. ^ Jacobs, Bart (1999). Categorical logic and type theory. Amsterdam Lausanne New York [etc.]: Elsevier. ISBN 0444501703.
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