In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more precisely, in the theory of
simplicial set
In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs.
Every simplicial set gives rise to a "n ...
s, a simplicial group is a
simplicial object
In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs.
Every simplicial set gives rise to a "n ...
in the
category of groups
In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.
Relation to other categories
The ...
. Similarly, a simplicial abelian group is a simplicial object in the
category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object o ...
. A simplicial group is a
Kan complex
In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are ...
(in particular, its
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s make sense). The
Dold–Kan correspondence
In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the ...
says that a simplicial abelian group may be identified with a
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...
. In fact it can be shown that
any simplicial abelian group
is non-canonically
homotopy equivalent
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
to a product of
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ...
s,
A
commutative monoid
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being .
Monoids are semigroups with identity ...
in the
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
of simplicial abelian groups is a
simplicial commutative ring
In algebra, a simplicial commutative ring is a monoid object, commutative monoid in the category (mathematics), category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If ''A'' is a simplic ...
.
discusses a simplicial analogue of the fact that a
cohomology class
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
on a
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...
has a unique
harmonic representative and deduces
Kirchhoff's circuit laws
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirc ...
from these observations.
See also
*
Simplicial commutative ring
In algebra, a simplicial commutative ring is a monoid object, commutative monoid in the category (mathematics), category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If ''A'' is a simplic ...
References
*
*
*
Charles Weibel
Charles Alexander Weibel (born October 28, 1950, in Terre Haute, Indiana) is an American mathematician working on algebraic K-theory, algebraic geometry and homological algebra.
Weibel studied physics and mathematics at the University of Michiga ...
, ''An introduction to homological algebra''
Further reading
*
What is a simplicial commutative ring from the point of view of homotopy theory?* W. G. Dwyer and D. M. Kan. “Homotopy theory and simplicial groupoids”. In: Nederl. Akad. Wetensch. Indag. Math. 46.4 (1984), pp. 379–385.
* http://pantodon.jp/index.rb?body=simplicial_group in Japanese
Simplicial sets
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