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Simplicial object in a category

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A contravariant functor (or, equivalently, a covariant functor ) from the category , whose objects are ordered sets , , and whose morphisms are non-decreasing mappings , into the category . A covariant functor (or, equivalently, a contravariant functor ) is called a co-simplicial object in .

The morphisms

of given by

generate all the morphisms of , so that a simplicial object is determined by the objects , (called the -fibres or -components of the simplicial object ), and the morphisms

(called boundary operators and degeneracy operators, respectively). In case is a category of structured sets, the elements of are usually called the -dimensional simplices of . The mappings and satisfy the relations

(*)

and any relation between these mappings is a consequence of the relations (*). This means that a simplicial object can be identified with a system of objects , , of and morphisms and , , satisfying the relations

Similarly, a co-simplicial object can be identified with a system of objects , (-co-fibres) and morphisms , (co-boundary operators), and , (co-degeneracy operators), satisfying the relations (*) (with , ).

A simplicial mapping between simplicial objects (in the same category ) is a transformation (morphism) of functors from into , that is, a family of morphisms , , of such that

The simplicial objects of and their simplicial mappings form a category, denoted by .

A simplicial homotopy between two simplicial mappings between simplicial objects in a category is a family of morphisms , , of such that

On the basis of this definition one can reproduce in essence the whole of ordinary homotopy theory in the category , for any category . In the case of the category of sets or topological spaces, the geometric realization functor (see Simplicial set) carries this "simplicial" theory into the usual one.

Examples of simplicial objects are a simplicial set, a simplicial topological space, a simplicial algebraic variety, a simplicial group, a simplicial Abelian group, a simplicial Lie algebra, a simplicial smooth manifold, etc.

Every simplicial Abelian group can be made into a chain complex with boundary operator .

References

[1] P. Gabriel, M. Zisman, "Calculus of fractions and homotopy theory" , Springer (1967)
[2] J.P. May, "Simplicial objects in algebraic topology" , v. Nostrand (1967)
[3] K. Lamotke, "Semisimpliziale algebraische Topologie" , Springer (1968)
How to Cite This Entry:
Simplicial object in a category. S.N. MalyginM.M. Postnikov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Simplicial_object_in_a_category&oldid=12531
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098