Let $k$ be commutative ring and $(C, \Delta)$ be a coalgebra over $k$. Let $D$ be a $k$-submodule of $C$.


Notes I'm reading give the following definition:



> 
> $D$ is called subcoalgebra of $C$ if the comultiplication $\Delta: C
>  \to C \otimes C$ restricts to a mapping $$\Delta\vert \_D: D \to D
>  \otimes D$$
> 
> 
> 


However, this seems a bit odd to me. Indeed, in general the inclusion $i: D \otimes D \to C \otimes C$ need not be injective so it might not be possible to view $D\otimes D \subseteq C \otimes C$.


What would be the appropriate definition in this context? I see several options:


(1) Assume that we work with flat modules (for example, vector spaces) so that the above makes sense. This seems too restrictive.


(2) Ask that $D$ itself is a coalgebra such that the inclusion $j: D \hookrightarrow C$ is a morphism of coalgebras.


Which definition (if any of these) is the one I should use?



Definition of subcoalgebra over a commutative ring
I'm surprised nobody has answered yet so let me state (what seems to me to be) the obvious: (2) is the only reasonable definition of sub-coalgebra which makes sense in general. Said differently you want $\Delta:D\rightarrow C\otimes C$ to factor through $D\otimes D \rightarrow C\otimes C$. The definition in your notes is either an oversight or an abuse of language, I'd say.


As მამუკა ჯიბლაძე suggests, coalgebras in $R$-mod are the same as algebras in $R-mod^{op}$, and you don't require the latter map to be injective in the same way that for a quotient algebra $A\rightarrow B$ you don't usually require $A\otimes A\rightarrow B\otimes B$ to be surjective as well (why would you ?). Of course for algebras over a commutative ring this is automatically true, but it's not true for symmetric monoidal categories whose tensor product is not right exact, which is exactly what happens for $R-mod^{op}$.


The downside is that the coalgebra structure on a sub-coalgebra might not be uniquely determined, i.e. it is possible to have two different coalgebra structures on $D$ for which the inclusion $D\rightarrow C$ is a coalgebra map. In other words, the factorisation of the map $D\rightarrow C\otimes C$ might not be unique. Somewhat related, if $C$ is not flat then its category of comodules might not be abelian.


See e.g. Wischnewsky, [On linear representations of affine groups. I.](https://projecteuclid.org/euclid.pjm/1102868049) for a reference where this is carefully phrased.


