Pro-simplicial set

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In mathematics, a pro-simplicial set is an inverse system of simplicial sets.

A pro-simplicial set is called pro-finite if each term of the inverse system of simplicial sets has finite homotopy groups.

Pro-simplicial sets show up in shape theory, in the study of localization and completion in homotopy theory, and in the study of homotopy properties of schemes (e.g. étale homotopy theory).


  • Edwards, David A.; Hastings, Harold M. (1980), "Čech theory: its past, present, and future", The Rocky Mountain Journal of Mathematics, 10 (3): 429–468, doi:10.1216/RMJ-1980-10-3-429, MR 0590209 .
  • Edwards, David A.; Hastings, Harold M. (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Mathematics, Vol. 542, Springer-Verlag, Berlin-New York, MR 0428322 .