symmetric monoidal (∞,1)-category of spectra
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
and a unit
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
and a counit
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
a general coassociative coalgebra is a coalgebra over a comonad, dual to the concept of an algebra over a monad.
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.
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.)
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
$\Sigma_i\xi_i\otimes v^{2^i}$
(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).
Moss Sweedler, Hopf algebras, 1969
Walter Michaelis, Coassociative Coalgebras, Handbook of Algebra Volume 3, Elsevier (2003).
Lowell Abrams, Charles Weibel, Cotensor products of modules (arXiv:math/9912211)
Ezra Getzler, Paul Goerss, A model category structure for differential graded coalgebras, 1999 (ps)
Kathryn Hess, Brooke Shipley, The homotopy theory of coalgebras over a comonad (arXiv:1205.3979)