Pro-simplicial set
Encyclopedia
In mathematics
, a pro-simplicial set is an inverse system
of simplicial set
s.
A pro-simplicial set is called pro-finite if each term of the inverse system of simplicial set
s 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).
References
1. DAVID A. EDWARDS AND HAROLD M. HASTINGS, CECH THEORY: ITS PAST, PRESENT, AND FUTURE,ROCKY MOUNTAIN
JOURNAL OF MATHEMATICS, Volume 10, Number 3, Summer 1980
http://rmmc.asu.edu/TO%20DOUGLAS/RMJ/vol10/vol10-3/edw.pdf
2. D.A. Edwards and H. M. Hastings, (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag.
http://bib.tiera.ru/dvd56/Edwards%20D.%20A.,%20Hastings%20H.%20M.%20-%20Cech%20and%20Steenrod%20Homotopy%20Theories%20with%20Applications%20to%20Geometric%20Topology%281976%29%28308%29.pdf
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a pro-simplicial set is an inverse system
Inverse system
In mathematics, an inverse system in a category C is a functor from a small cofiltered category I to C. An inverse system is sometimes called a pro-object in C. The dual concept is a direct system.-The category of inverse systems:...
of simplicial set
Simplicial set
In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space...
s.
A pro-simplicial set is called pro-finite if each term of the inverse system of simplicial set
Simplicial set
In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space...
s has finite
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
homotopy groups.
Pro-simplicial sets show up in shape theory
Shape theory (mathematics)
Shape theory is a branch of the mathematical field of topology. Homotopy theory is not appropriate for spaces with bad local properties, hence the need for replacement of homotopy theory by a more sophisticated approach...
, in the study of localization and completion in homotopy theory, and in the study of homotopy properties of schemes (e.g. étale homotopy theory).
References
1. DAVID A. EDWARDS AND HAROLD M. HASTINGS, CECH THEORY: ITS PAST, PRESENT, AND FUTURE,ROCKY MOUNTAIN
JOURNAL OF MATHEMATICS, Volume 10, Number 3, Summer 1980
http://rmmc.asu.edu/TO%20DOUGLAS/RMJ/vol10/vol10-3/edw.pdf
2. D.A. Edwards and H. M. Hastings, (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag.
http://bib.tiera.ru/dvd56/Edwards%20D.%20A.,%20Hastings%20H.%20M.%20-%20Cech%20and%20Steenrod%20Homotopy%20Theories%20with%20Applications%20to%20Geometric%20Topology%281976%29%28308%29.pdf