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mapping a terminal object to a coproduct of terminal objects

Step-by-step mathematics solution: mapping a terminal object to a coproduct of terminal objects

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1. What the problem is asking

We work in a category (\mathcal C) that

  • has a terminal object (1);
  • has finite coproducts (\;\coprod).

For a fixed integer (n\ge 1) we form the coproduct of (n) copies of the terminal object

[ \underbrace{1\;\coprod\;1\;\coprod\;\cdots\;\coprod\;1}_{n\text{ times}}\;=\;\coprod^{\,n}1 . ]

Every coproduct comes equipped with injection maps

[ \iota_i : 1 \longrightarrow \coprod^{\,n}1 ,\qquad i=1,\dots ,n . ]

The question is:

When does the set of all arrows (1\to\coprod^{\,n}1) consist exactly of those (n) injections?

In other words, for which categories is the canonical function

[ \Phi_n \;:\; {1,\dots ,n}\;\longrightarrow\;\operatorname{Hom}_{\mathcal C}!\bigl(1,\coprod^{\,n}1\bigr),\qquad i\mapsto \iota_i\circ!(!_1) ]

a bijection for every (n\ge 1)?

The author calls this the “one‑to‑(n) property”.
We have to relate it to known categorical notions and give a clean characterisation.

2. A systematic analysis

2.1 The canonical map (\Phi_n)

Because (1) is terminal, there is a unique map (!_{1}:1\to 1).
Hence each injection (\iota_i) gives a global element (arrow from the terminal object)

[ \iota_i\circ !_1 : 1\longrightarrow\coprod^{\,n}1 . ]

Collecting them gives a function

[ \Phi_n : n \;=\;\coprod_{i=1}^{n}\operatorname{Hom}(1,1) \;\longrightarrow\;\operatorname{Hom}\bigl(1,\coprod^{\,n}1\bigr). \tag{1} ]

The domain has exactly (n) elements because (\operatorname{Hom}(1,1)={!_{1}}).

Injectivity of (\Phi_n) is automatic in any category: the injections (\iota_i) are distinct as morphisms of the coproduct, so the corresponding global elements are distinct.
The hard part is surjectivity: we must show that every global element of (\coprod^{\,n}1) comes from one of the injections.

Thus the “one‑to‑(n) property’’ is exactly the statement

[ \boxed{\text{For every }n,\; \Phi_n \text{ is a bijection}.} \tag{2} ]

2.2 Interpreting (\Phi_n) as preservation of coproducts

Define the global‑sections functor

[ \Gamma\;=\;\operatorname{Hom}_{\mathcal C}(1,-)\;:\;\mathcal C\longrightarrow \mathbf{Set}. ]

For any two objects (A,B) we have a natural map

[ \Gamma A\;\amalg\;\Gamma B \;\longrightarrow\;\Gamma(A\amalg B) ]

obtained by composing with the coproduct injections.
When the source objects are both the terminal object we obtain precisely (\Phi_n) :

[ \Gamma(1)\amalg\cdots\amalg\Gamma(1) \;\cong\; n \;\xrightarrow{\;\Phi_n\;}\; \Gamma!\bigl(\coprod^{\,n}1\bigr). ]

Consequently,

[ \text{(2) holds for all }n \iff \Gamma \text{ preserves the coproduct }\coprod^{\,n}1. ]

Because (\Gamma) already preserves the empty coproduct (it sends the initial object to the empty set), condition (2) is equivalent to

[ \boxed{\Gamma\text{ preserves all finite coproducts}.} \tag{3} ]

Thus the “one‑to‑(n) property’’ is nothing other than the statement that global sections turn a coproduct of copies of the terminal object into the ordinary set‑theoretic coproduct of the corresponding copies of a one‑point set.

2.3 Relation to familiar categorical notions

  1. Well‑pointedness (in topos theory).
    For an elementary topos (\mathcal E) the following are equivalent

    • (i) (\mathcal E) is well‑pointed, i.e. the terminal object (1) is a separator (a generator) and the global‑sections functor (\Gamma) is faithful;

    • (ii) (\Gamma) preserves finite coproducts.

    In a well‑pointed topos the object (\coprod^{\,n}1) is the discrete object with exactly (n) points, so the one‑to‑(n) property holds.

  2. Generator + preservation of coproducts.
    In an arbitrary category (not necessarily a topos) condition (3) can be split:

    • (1) being a generator (or separator) guarantees that the map (\Phi_n) is injective (different injections give different arrows).

    • Preservation of coproducts guarantees surjectivity.

    Hence the one‑to‑(n) property holds exactly when (1) is a generator and (\Gamma) preserves finite coproducts.

  3. Extensive categories.
    In an extensive category coproduct injections are disjoint and universal, but this does not force (3). The total order with a top element is extensive (its coproducts are joins), yet (\Gamma) does not preserve them, and the property fails – exactly the example mentioned in the question.

  4. Free coproduct completion.
    The free coproduct completion (\operatorname{Fam}(\mathcal C)) of any category (\mathcal C) has the property, because objects are families ((X_i)_{i\in I}) and a map (1\to\coprod^{\,n}1) is just the choice of an index (i); therefore (\Gamma) sends (\coprod^{\,n}1) to the set ({1,\dots ,n}). This fits the general characterisation: (\Gamma) preserves coproducts in (\operatorname{

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