TITLE: What is a most elementary Coalgebra?
QUESTION [13 upvotes]: I just read the definition of a coalgebra, defined categorically ( reversing the arrows of the Algebra category ), the given example in the text I am referring to is the Homology on a topological space. I am not familiar yet with algebraic topology, I barely recall something about homotopy groups.
So I am trying to come up with simpler examples, hence the questions: 
What is an elementary example of a Coalgebra ? (if any) 
How would you define it without using the language of Category ? i.e. for a set $X$ what are the axioms needed to tell that $(X,+,\times)$ has a Coalgebra structure ? 
well here $(X,+)$ must be a $\mathbb{K}$-module as in an Algebra, but what to say about the comultiplication $\times$ ? 
( and why do we call it co- , and why it is the dual of the multiplication )
This push me to think about how to transform any Categorical definition to the axiomatic definition...

REPLY [4 votes]: The simplest examples arise from free modules with chosen bases (over a commutative ring; for a non-commutative ring you have to use bimodules to get the tensor product to work).
Explicitly, the tensor product $M\otimes M$ of a a free $R$-module $M$ with chosen basis $X\subseteq M$ has basis $\{x_1\otimes x_2:x_1,x_2\in X\}\cong X\times X$. It turns out that the choice of basis $X$ for $M$ gives a coalgebra structure with comultiplication the linear map $M\to M\otimes M$ generated by $x\mapsto x\otimes x$ and that has a counit the linear map $M\to R$ generated by $x\mapsto 1$. More explcitly, the comultiplication is given by $\sum_ir_ix_i\mapsto\sum r_ix_i\otimes x_i$, and the counit by $\sum_i r_ix_i\mapsto\sum_ir_i$.
The next simplest examples arise from free modules of finite rank (i.e. admitting a finite basis) with an algebra structure. Namely, given an $R$-algebra $M$ with multiplication $m\colon M\otimes M\to M$ and unit $e\colon R\to M$, these correspond to an $R$-linear map $m^\vee\colon M^\vee\to(M\otimes M)^\vee$ and an $R$-linear map $e^\vee\colon M^\vee\to R$, where $M^\vee=\{f\colon M\to R\colon f$ is $R$-linear$\}$. In the case $M$ is free of finite rank, we have an isomorphism $(M\otimes M)^\vee\cong M^\vee\otimes M^\vee$, so a (unital) algebra structure on $M$ induces a coalgebra (with counit) structure on $M^\vee$, the $R$-module dual to $M$.