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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simplicial_object_in_a_category&oldid=49586
This article was adapted from an original article by S.N. MalyginM.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
