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This is more a question about notation and definitions. I have been looking a bit at Grothendieck's Tohoku paper, and one of the important constructions in an abelian category is that of the categorical intersection. Let $\mathcal{A}$ be an abelian category. My understanding is that if you have an object $A$ of $\mathcal{A}$ with a family of subobjects $\{ \mu_{i}: A_{i} \hookrightarrow A \}$ (where the $\mu_{i}$ are monomorphisms) then the categorical intersection $\cap A_{i}$ is simply the pullback of these. Since a pullback of monomorphisms is a monomorphism, we recover a subobject of $A$. Is this correct?

I am more confused about the definition of the union. First of all, is it true that the categorical sum $\Sigma A_{i}$ and categorical union $\cup{A_{i}}$ are just different names for the same thing? Assuming they are, I will use the notation $\Sigma A_{i}$. My understanding is that in a cocomplete abelian category, the categorical union is just given by the image of the morphism $$ \bigoplus_{i} A_{i} \longrightarrow A. $$ Is this correct? In the case of a Grothendieck abelian category (or indeed just an AB4 abelian category), we have cocompleteness and the fact that the above direct sum of monomorphisms is a monomorphism itself, so the direct sum is isomorphic to the image. In that case, surely we can just say that the categorical union is precisely that direct sum? But it seems like no texts actually say that. Why introduce all this new notation, such as sigma or $\cup$ if it is merely a direct sum of subobjects? Why not just call it that? Or have I massively misunderstood something?

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  • $\begingroup$ I don't know what your definitions of "categorical sum" and "categorical union" are, but I can't imagine how they could be different in this context. $\endgroup$ – Eric Wofsey Dec 10 '17 at 2:52
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Your understanding is correct except that you have a massive misunderstanding of what "direct sum of monomorphisms is a monomorphism" means. It means that if you have a family of monomorphisms $A_i\to B_i$, then the induced map $\bigoplus A_i\to\bigoplus B_i$ is a monomorphism. It does not mean that if you have monomorphisms $A_i\to B$, the induced map $\bigoplus A_i\to B$ is a monomorphism. Indeed, this is rarely true.

As is almost always the case, your first instinct to understand this should be to consider simple examples in a category of modules over a ring. Here's a very simple example: in $Ab$, consider the subobjects $2\mathbb{Z}\to\mathbb{Z}$ and $3\mathbb{Z}\to\mathbb{Z}$. The induced map $2\mathbb{Z}\oplus3\mathbb{Z}\to\mathbb{Z}$ is not injective (its kernel is generated by $(6,-6)$), but its image is $\mathbb{Z}$, what is usually called the sum of the subgroups $2\mathbb{Z}$ and $3\mathbb{Z}$.

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