Do coalgebras arise outside the study of bi/Hopf-algebras?

Question

Hopefully the title is fairly self explanatory. I'm curious as to whether the coalgebra structure (that is, a vector space with a comultiplication and counit) comes up an any area of mathematics not specifically because it is the arrow reversal of a unital algebra.

Answer 1

If $X$ is a set, then the canonical duplication $X \to X \times X$ map gives a coalgebra on $\mathbb{K}[X]$, the vector space with basis $X$. This is because $\mathbb{K}[X \times X] \cong \mathbb{K}[X] \otimes \mathbb{K}[X]$.

The induced comultiplication $\mathbb{K}[X] \to \mathbb{K}[X] \otimes \mathbb{K}[X]$ can be used to define a Hopf-algebra structure on $\mathbb{K}[G]$ for $G$ a finite group.

There is also another coalgebra on $\mathbb{C}[G]$ the group ring.

$\delta : \mathbb{C}[G] \to \mathbb{C}[G] \otimes \mathbb{C}[G]$ "unmultiplies" an element:

$\delta (g) = \frac{1}{|G|}\sum_{h \in G} gh^{-1} \otimes h$

Answer 2

Here are two other examples where one is lead to consider coalgebras by themselves, that are not necessarily Hopf algebras.

The singular chain complex $\mathcal{S}_*(X)$ of a topological space gas a coalgebra structure, by the Eilenberg–Zilber theorem. This is also an instance of the codiagonal $X \to X \times X$ appearing. If you work with field coefficients, this gives a coalgebra structure on the homology $H_*(X; \Bbbk)$ of the space. (The field coefficients are here to deal with the fact that the Künneth morphism goes the wrong way; see this MO Q&A for more details). If you want to get a Hopf algebra structure on homology you probably, need some kind of H-space structure on $X$.

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In homological algebra, one is often interested in constructing resolution of algebras. Given an (augmented) algebra $A$, its bar construction $BA = (T^c(\Sigma A), b)$ is a coalgebra. If $A$ is not commutative then there is no (to my knowledge) natural Hopf algebra structure on $BA$ to consider.
Then the cobar construction $\Omega BA$ is an algebra which is a resolution of $A$: $\Omega BA \xrightarrow{\sim} A$ (see eg. the $n$Lab article for some pointers).

Given a quadratic algebra $A = T(V)/(S)$, one can first consider its Koszul dual algebra $A^! = T(V^*)/(S^\perp)$, but it's also interesting to consider its Koszul dual coalgebra $A^¡ = T^c(V, S)$, cogenerated^(*) by $V$ with corelations $S$. There is a natural map of algebras $\Omega A^¡ \to A$, and this map is a resolution when the algebra is Koszul. In this case it gives a very interesting, "small" resolution of $A$.

More generally, you can consider quadratic operads (in some sense, operads are generalizations of unital algebras). Given such a quadratic operad $\mathtt{P}$, you can construct its Koszul dual cooperad $\mathtt{P}^¡$ (in the same sense a cooperad is a generalization of a coalgebra). When the operad is Koszul you get a resolution $\Omega \mathtt{P}^¡ \xrightarrow{\sim} \mathtt{P}$. See eg. the book Algebraic Operads of Loday and Vallette.
Even more generally, you can consider monogenic algebras over a quadratic operad, and this leads one to consider Koszul dual coalgebras over some quadratic cooperad. See Millès, "The Koszul complex is the cotangent complex". So for example you would see Lie coalgebras, cocommutative coalgebras, Poisson coalgebras... appear. In this case it might not even be necessarily clear what the correct notion of "Hopf algebra" would be.

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^(*) There are actually suspensions involved but I don't want to get bogged down in technical details. In fact most of the last paragraph is very imprecise.

Answer 3

Another area, where the ideas of coalgebras were implicitly present, even before the formal definitions were laid, is the study of the representations of Lie groups and Lie algebras: For example, if $L$ is a Lie algebra and ${}_LM$, ${}_LN$ are two $L$-modules, then their tensor product $M\otimes N$ becomes also an $L$-module through ($l\in L$) $$l\cdot\big(m\otimes n\big)=\Delta(l)\cdot (m\otimes n)=(l\cdot m)\otimes n+m\otimes(l\cdot n)$$ Similarly, group $G$ representations can be tensored through the group comultiplication ($g\in G$): $$g\cdot\big(m\otimes n\big)=\Delta(g)\cdot(m\otimes n)=(g\cdot m)\otimes(g\cdot n)$$ I actually think, that such remarks can be classified among the motivating remarks which lead to the ideas of comultiplication, coactions, etc