When is coproduct called disjoint union

category-theorycoproductterminology

Disjiont union forms coproduct in the category of sets. The coproduct in the category of topological spaces is given by endowing a natural topology on the disjiont union of the underlying sets of spaces, and is sometimes called the disjoint union of spaces. Can I call coproduct disjiont union in similar cases?

Let $F\colon\mathcal C\to\mathrm{Set}$ be a forgetful functor of a category $\mathcal C$ into the category of sets. If $F$ preserves coproduct, is it normal to refer to coproduct in $\mathcal C$ as disjiont union? This is the most general setting I can think of and I also wonder if there is a more general one.

Best Answer

Suppose given an object $X$ and morphisms $i_a : X_a \to X$ for each $a$ in some index set $A$. These comprise a disjoint coproduct if the following conditions are satisfied:

  • Given $a$ and $a'$ in $A$ and morphisms $x : T \to X_a$ and $x' : T \to X_{a'}$, we have $i_a \circ x = i_{a'} \circ x'$ if and only if

    • $a = a'$ and $x = x'$, or
    • $x$ and $x'$ factor through an initial object.

  • $i_a : X_a \to X$ comprise a coproduct cocone, i.e. for every object $Y$, given morphisms $f_a : X_a \to Y$ for each $a \in A$, there is a unique morphism $f : X \to Y$ such that $f \circ i_a = f_a$ for all $a \in A$.

This definition is a little bit more general than the definition appearing in nLab. More precisely, supposing we have a disjoint coproduct as above:

Proposition 1. For each $a \in A$, $i_a : X_a \to X$ is a monomorphism.

Proposition 2. Given $a$ and $a'$ in $A$ and an initial object $0$:

  • $a = a'$, or
  • we have a pullback diagram of the form below: $$\require{AMScd} \begin{CD} 0 @>>> X_{a'} \\ @VVV @VV{i_{a'}}V \\ X_a @>>{i_a}> X \end{CD}$$

Proposition 3. Given $a$ and $a'$ in $A$ and a pullback square of the form below, $$\begin{CD} U @>{x'}>> X_{a'} \\ @V{x}VV @VV{i_{a'}}V \\ X_a @>>{i_a}> X \end{CD}$$ then:

  • $a = a'$ and $x = x'$ is an isomorphism, or

  • $U$ is an initial object.

So in a category that has an initial object, the definition above is equivalent to the usual one. The empty coproduct and the unary coproduct are automatically disjoint, but in general even binary coproducts can fail to be disjoint. However, the definition is a bit too general, in the sense that even coproducts in $\textbf{Ab}$ and $\textbf{Top}_*$ are disjoint in this sense. One important feature of coproducts in $\textbf{Set}$ and $\textbf{Top}$ is that any morphism into a coproduct induces a coproduct decomposition of the domain. Categories where coproducts (exist and) have this property are said to be extensive, and in extensive categories coproducts are disjoint.

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