Grothendieck construction

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The Grothendieck construction (named after Alexander Grothendieck) is a construction used in the mathematical field of category theory.


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 .


"The Grothendieck construction takes structured, tabulated data and flattens it by throwing it all into one big space. The projection functor is then tasked with remembering which box each datum originally came from." [1]


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[edit]


  • 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.
  1. ^ 1978-, Spivak, David I.,. Category theory for the sciences. Cambridge, Massachusetts. pp. ISBN 0262028131. OCLC 893909845.

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