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Definition

There are two different meanings of the term coefficient in mathematics, one

and one

The two notion do however coincide for ordinary homology/ordinary cohomology expressed as singular homology/singular cohomology. See remark below.

In algebra and analysis

In algebra and analysis, a coefficient is an element of a ring (or rig) $R$ that appears in scalar multiplication; more generally, coefficients are elements of $R$ that appear in a linear combinations. Thus, we multiply a coefficient by an element of an $R$-module $M$ (which may even be an algebra, associative or not) to get another element of $M$.

For example, given a polynomial $P = \sum_n a_n x^n$ over $R$ in the variable $x$, each $a_n \in R$ is the coefficient on $x^n$ in $P$. The module $M$ here is the symmetric algebra over $R$.

In cohomology and homology

In cohomology coefficients are what the cohomology takes values in. For ordinary cohomology $H^\bullet(-,A)$ the abelian group $A$ is the coefficient group. For generalized (Eilenberg-Steenrod) cohomology $H^\bullet(-,E)$ the given spectrum $E$ that represents it is the coefficient spectrum. Dually for homology.

It is this notion of “coefficient” that appears in terms like

Remark

This cohomology-theoretic usage of “coefficient” is almost certainly originally derived from the former one. When ordinary homology with coefficients in $A$ (in the homological sense) is defined using singular or cellular chains, the chains themselves are formal linear combinations of singular simplices or cells whose coefficients (in the algebraic sense) are elements of $A$. For more general homology and cohomology theories, however, the “coefficient object” is no longer directly interpretable using coefficients in the algebraic sense.

category: disambiguation