# nLab coalgebra

Coalgebra

### Context

#### Algebra

higher algebra

universal algebra

# Coalgebra

## Definition

In the most familiar sense, a coalgebra is just like an associative algebra, but with all the structure maps ‘turned around’ (a “co-monoid”). More precisely, fix a ground field $k$. Then an associative algebra $A$ over $k$ is a vector space equipped with a multiplication

$m : A \otimes A \to A$

and a unit

$i : k \to A$

satisfying the associativity law and left/right unit laws, which can be drawn as commutative diagrams. Similarly, a coalgebra $C$ is a vector space equipped with a comultiplication

$\Delta : A \to A \otimes A$

and a counit

$e: A \to k$

satisfying the coassociative and left/right counit laws. The commutative diagrams for these laws are obtained by taking the diagrams for the associative and left/right unit laws and turning all the arrows around.

We can express this idea much more efficiently using the concept of the opposite of a category, together with internalization. Namely: a coalgebra is a monoid in the $Vect^{op}$, just as an algebra is a monoid in $Vect$.

Coalgebras of this sort are an important ingredient in more sophisticated structures such as bialgebras, Hopf algebras and Frobenius algebras.

More generally:

• a coalgebra for an endofunctor $F : C \to C$ on a category $C$ – an $F$-coalgebra – is

• an object $A$ of $C$;

• and a morphism $\alpha : A \to F(A)$;

• a general coassociative coalgebra is a coalgebra over a comonad, dual to the concept of an algebra over a monad.

## Examples

### Special cases

• For $R$ a commutative ring, if the endofunction $F : C \to C$ is $F : R Mod \to R Mod$ given by $F : N \mapsto N \otimes N$, then $F$-coalgebras are precisely non-coassociative coalgebras in the specific sense of non-associative monoids in $R Mod^{op}$. (See Tom Leinster‘s comment here).

• $L_\infty$-algebras are cocommutative comonoids in the category of chain complexes.

### Other

These are explored briefly in the lexicon style entry differential graded coalgebra. (At present this is ‘bare bones’ with little or no motivation or discussion.)

## Cocommutative coassociative coalgebras

These, in most cases, form a complete cocomplete Cartesian Closed Category, $Coalg$ over which the category, $Alg$, of commutative associative algebras is enriched, tensored and cotensored. The exegesis is much the same whether we consider coalgebras over a field $k$, or graded $k$-coalgebras, or differential graded coalgebras, etc. In each case we need a notion of finiteness: finite $k$-dimension of the underlying $k$-vector space, finite dimension in each grade, etc. We denote by $Alg_f$ the category of (commutative associative) algebras that are finite.

The basic fact is that a coalgebra is the filtered colimit of its finite dimensional subcoalgebras. It follows from this that we can identify $Coalg$ with the category of finite-limit-preserving $Set$-valued functors on $Alg_f$. This is because every such functor is a filtered colimit of representable functors, and for any finite algebra $A$ its $k$-vector-space dual $A^*$ is a finite coalgebra.

The product of coalgebras $C$ and $D$ is given by $C\otimes_k D$. The exponential $C\Rightarrow D$ is given by the functor taking $A\in Alg_f$ to $Hom_{Coalg}(A^*\otimes_k C,D)$. Note that $C\Rightarrow C$ has the structure of a cocommutative coassociative Hopf algebra.

For $X$ and $Y$ in $Alg$ we define $X\Rightarrow Y$ in $Coalg$ to be given by the functor taking $A\in Alg_f$ to $Hom_{Alg}(X,A\otimes_k Y)$.

For $C\in Coalg$ and $X\in Alg$ we denote by $C\Rightarrow X$ the algebra $Hom_k(C,X)$ with $k$-algebra structure induced by the coalgebra structure of $C$. We denote by $C\otimes X$ the quotient of the free $k$-algebra on $C\otimes_k X$ by the ideal generated by elements of the form

$c\otimes 1-\epsilon(c)$ and $c\otimes x_1 x_2 - \Sigma_i (c'_i\otimes x_1)(c''_i\otimes x_2)$

where $\epsilon$ is the counit of $C$ and $\Sigma_i c'_i\otimes c''_i$ is the diagonal of $c$ in $C$.

The tensored, cotensored enrichment of $Alg$ over $Coalg$ can be extended to the case of commutative associative $k$-algebras in a topos. It is a consequence of work by N.J.Kuhn, Generic representations of the Finite General Linear groups and the Steenrod Algebra, that the mod 2 Steenrod algebra is the Hopf algebra $S(V)\Rightarrow S(V)$ where $S(V)$ is the free graded symmetric $\mathbb{Z}_2$-algebra on the generic $\mathbb{Z}_2$-vectorspace. Similar considerations apply to the mod p Steenrod algebra.

Recall that the mod 2 Steenrod Hopf algebra is the dual of the commutative Hopf algebra with generators $\xi_1, \ldots$ with diagonal taking $\xi_n$ to

$\Sigma_{i+j=n} \xi_i^{2^j}\otimes \xi_j$

where $\xi_0 = 1$. Its action on $S(V)$ is dual to the coaction taking a vector $v\in V$ to

$\Sigma_i\xi_i\otimes v^{2^i}$

## Properties

### As filtered colimits of finite-dimensional pieces

###### Theorem

(fundamental theorem of coalgebras)

Every coalgebra is the filtered colimit of its finite-dimensional sub-coalgebras.

This is maybe due to (Sweedler 69), for proof see also for instance (Michaelis 03). That this remains true for dg-coalgebras (see at dg-coalgebra – As filtered colimit) is due to (Getzler-Goerss 99).

###### Remark

It follows that an algebra, while not itself the filtered limit of its finite dimensional subalgebras in general, is, being the linear dual of a coalgebra, a “formal filtered limit”, hence a pro-object in finite-dimensional algebras (e.g. Abrams-Weibel 99, p. 7).

## References

Last revised on May 3, 2022 at 20:36:35. See the history of this page for a list of all contributions to it.