"Partition" of an object in a category? Is there a reasonable and useful way of defining a partition of an object in a category the way a partition of a set is defined? I have not found such a definition anywhere so far.
My first thought was to define a partition of an object $X$ as a collection of monomorphisms $A_i\to X$ such that $X$ is isomorphic the coproduct of the $A_i$ and the pullback of each of the diagrams $A_i\to X\leftarrow A_j$ $(i\ne j)$ is the initial object.
In the category of sets, a partition of a set is a particular case of this construction (I think...)
But  we also find this notion in other categories. I believe that in the category of vector spaces, a direct sum decomposition also gives a "partition" in this sense. Since I am obviously not the first person to ever think of this, could someone please provide a reference where this (or a similar) construction is explored?
 A: Lawvere and Rosebrugh, in their book "Sets for Mathematics", define partitions simply as epimorphisms.  In the category of sets, epimorphisms coincide with surjections, and a surjection can be expressed as the partition of its fibers.  On the flip side, every partition gives a quotient projection which is surjective.
A: The first part seems to be related to the negation of the property of indecomposable objects in a category.
I guess that you included the second part wishing that it stood for the subobjects being "disjoint" in a sense, but in fact that condition does not capture the appropiate kind of disjointness e.g. for vector spaces. There you would have to ask that, for every $i$, the pullback of $A_i\to X\leftarrow \coprod_{j\neq i}A_j$ is the initial object.
This stronger version of your second condition is automatically fulfilled for coproducts in the categories of sets, topological spaces, differentiable manifolds and even in the categories of modules over a ring, but it is not in the case of rings, where e.g. the diagram $\mathbb Q\to\mathbb Q\leftarrow \mathbb Q$ is a coproduct sink, so I think the stronger second condition is closer to what you mean.
Unfortunately, besides the nlab link I gave at the beggining, I don't know about a place where people discuss this. Interesting question.
