nLab coalgebra




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 kk. Then an associative algebra AA over kk is a vector space equipped with a multiplication

m:AAA m \colon A \otimes A \to A

and a unit

i:kA i \colon k \to A

satisfying the associativity law and left/right unit laws, which can be drawn as commutative diagrams.

Dually, a coalgebra CC is a vector space equipped with a comultiplication

Δ:AAA \Delta \colon A \to A \otimes A

and a counit

e:Ak e \colon A \to k

satisfying co-associativity and left/right co-unitality 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 opVect^{op}, just as an algebra is a monoid in VectVect.

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

More generally:


Special cases

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

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

Classes with extra properties and structure


Differential graded coalgebras

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, CoalgCoalg over which the category, AlgAlg, of commutative associative algebras is enriched, tensored and cotensored. The exegesis is much the same whether we consider coalgebras over a field kk, or graded kk-coalgebras, or differential graded coalgebras, etc. In each case we need a notion of finiteness: finite kk-dimension of the underlying kk-vector space, finite dimension in each grade, etc. We denote by Alg fAlg_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 CoalgCoalg with the category of finite-limit-preserving SetSet-valued functors on Alg fAlg_f. This is because every such functor is a filtered colimit of representable functors, and for any finite algebra AA its kk-vector-space dual A *A^* is a finite coalgebra.

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

For XX and YY in AlgAlg we define XYX\Rightarrow Y in CoalgCoalg to be given by the functor taking AAlg fA\in Alg_f to Hom Alg(X,A kY)Hom_{Alg}(X,A\otimes_k Y).

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

c1ϵ(c)c\otimes 1-\epsilon(c) and cx 1x 2Σ i(c ix 1)(c ix 2)c\otimes x_1 x_2 - \Sigma_i (c'_i\otimes x_1)(c''_i\otimes x_2)

where ϵ\epsilon is the counit of CC and Σ ic ic i\Sigma_i c'_i\otimes c''_i is the diagonal of cc in CC.

The tensored, cotensored enrichment of AlgAlg over CoalgCoalg can be extended to the case of commutative associative kk-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)S(V)S(V)\Rightarrow S(V) where S(V)S(V) is the free graded symmetric 2\mathbb{Z}_2-algebra on the generic 2\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 ξ 1,\xi_1, \ldots with diagonal taking ξ n\xi_n to

Σ i+j=nξ i 2 jξ j\Sigma_{i+j=n} \xi_i^{2^j}\otimes \xi_j

where ξ 0=1\xi_0 = 1. Its action on S(V)S(V) is dual to the coaction taking a vector vVv\in V to

Σ iξ iv 2 i\Sigma_i\xi_i\otimes v^{2^i}


As filtered colimits of finite-dimensional pieces


(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).


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).


On local presentability of categoris of coalgebras and their comodules:

Last revised on November 6, 2023 at 07:47:38. See the history of this page for a list of all contributions to it.