How to think about coalgebras? Are there geometric interpretations of coalgebras? 
If I think of algebras and modules as spaces and vectorbundles, what are coalgebras and comodules? What basic examples of coalgebras should one keep in mind?


Anything that helps to think about coalgebras without headache is welcome ;) 



What is a coalgebra intuitively?
It seems like there are two basic sources of examples:



1. 


The basic structure you have on a space (set, scheme...) is the diagonal morphism $\Delta :X \to X\times X$. Functions on spaces are contravariant, which is why functions on a space form an algebra: $f.g = \Delta ^\ast (f \boxtimes g)$. We also have the morphism $\pi : X \to pt$, and the unit in this algebra is $\pi ^\ast 1$.


If we chose some covariant linearization of our space (like measures, topological chains...) then this space would be a coalgebra, with comultiplication given by $\Delta\_\ast$, and counit $\pi\_\ast$. So, for example we have the coalgebra $C\_\ast (X)$ of (say, singular) chains on a topological space.


Such coalgebras are naturally cocommutative.


In nice cases, we have a pushforward and a pullback (e.g. functions on a finite set, equipped with a measure), and the algebra and coalgebra structures together form a *Frobenius algebra* (the inner product is $\pi \_\ast \Delta ^\ast$.



2. 


If our space $X$ is a group (or maybe just a monoid), then we have a multiplication map $m: X\times X \to X$, and then $m^\ast$ equips the space of functions (or, e.g. cochains) with the structure of a colagebra. If the multiplication on $X$ has a unit $e: pt \to X$, then $e^\ast$ is the counit of this algebra.


This coalgebra will not be cocommutative in general (unless $(X,m)$ is commutative).


If we also remember the ordinary multiplication structure (coming from $\Delta ^\ast$), then these two structures toegether form a *Hopf algebra* (also using the inversion $i:X \to X$ in the group).


Julian's example is of this second kind. 


All the examples I know, are morally either of type 1 or type 2, but sometimes you need to have a very broad definition of "function"or "measure"!


