Grothendieck construction

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The Grothendieck construction 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 .

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.

External links[edit]