# Definition of the Coproduct of Categories?

Let $\{\mathbf{C}_i~|~i \in I\}$ be a family of categories. What is $\coprod_{i\in I}\mathbf{C}_i$ ? Google and my regular sources have failed me. I might be able to anticipate (guess) the definition, but I would prefer not to. The setting I am working in is essentially small additive categories, but I don't think that matters.

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The coproduct of ordinary categories differs from the coproduct of $\mathbf{Ab}$-enriched categories, which also differs from the coproduct of additive categories. – Zhen Lin Jun 13 '13 at 7:10
In that case, I am looking for the definition involving additive categories. – Aaron McBride Jun 13 '13 at 7:29
In fact all colimits exist, see math.stackexchange.com/questions/272479 – Martin Brandenburg Jun 13 '13 at 9:23

Intuitions:

• Paste the "diagrams" representing the categories side-by-side;
• A book with $I$ pages: page $i$ contains $\mathbf C_i$, the book is $\coprod_{i\in I} \mathbf C_i$.

Formally:

• Objects: $\coprod_{i \in I}\operatorname{ob} \mathbf C_i$;
• Morphisms: $f: (i, C) \to (i,D)$ for $f: C \to D$ a morphism of $\mathbf C_i$

It is not hard to verify that this yields a category, nor is it hard to verify that it has the universal property of the coproduct in $\mathsf{Cat}$.

As to "guessing": it is a perfectly fine approach to try and guess what some limit/colimit should be. When trying to verify the universal property, you will either succeed, or learn something about why your construction does not work. In either case, you've learned. :)

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It's just the disjoint union of the categories. If the categories happen to be completely disjoint (meaning not having any object nor any arrow in common) then the coproduct of the family of categories is the category $C$ whose objects is the unions of all classes of objects $ob(C_i)$, and whose class of morphisms is similarly the union of all morphisms from the $C_i$. It is then obvious how to endow $C$ with a category structure, get the injections $C_i\to C$, and prove the universal property. If the categories were disjoint to begin with, change each one formally to obtain isomorphic copies which are disjoint. Then do the above construction on these copies.

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The Coproduct $\coprod_i C_i$ is given by $\def\Ob{\mathop{\rm Obj}\nolimits}\def\Mor{\mathop{\rm Mor}\nolimits}\Ob\coprod_i C_i = \bigcup_i \{i\} \times \Ob C_i$ for the objects, that is this disjoint union (=coproduct in $\mathbf{Set}$) and $$\Mor_{\coprod_i C_i}\bigl((i,A), (j,B)\bigr) = \begin{cases} \Mor_{C_i}(A,B) & i=j\\ \emptyset & i \ne j \end{cases}$$ The injection functors $J_i\colon C_i \to \coprod_i C_i$ are the obvious ones, on the object level given by the $\mathbf{Set}$ injections and on the morphisms the indentity on the morphism sets.

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