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Let $\mathbf{C}$ be a category with binary Cartesian products. Let's say that a map $f : X \to Y$ is constant if, for every pair of maps $g,h : A \to X$, maps $f \circ g$ and $f \circ h$ are equal. Now, we can define a "monoid object without points" in $\mathbf{C}$ as follows: it is an object $X$ together with a map $* : X \times X \to X$ and a constant map $e : X \to X$ such that $*$ is associative in the usual sense and both compositions $X \xrightarrow{\langle id, e \rangle} X \times X \overset{*}\to X$ and $X \xrightarrow{\langle e, id \rangle} X \times X \overset{*}\to X$ are equal to the identity morphism. We also can define a "group object without points" in a similar way.

If $\mathbf{C}$ has a terminal object and there is a map $x : 1 \to X$, then $X$ is a monoid object since we can define the neutral element as $1 \overset{x}\to X \overset{e}\to X$. Moreover, this defines a bijection between the set of monoid structures and "monoid without points" structures. In particular, the only "monoid objects without points" in $\mathbf{Set}$ are usual monoids and the empty set, but there might be more examples in an arbitrary category.

Is there a name for such "monoid" and "group" objects? Are there any interesting examples of them? Were they considered in the literature?

The reason I'm asking is that (I believe) I have an example of a locale which might not have points but has a structure of a group in the sense I described.

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    $\begingroup$ If the coequalizer $T$ of the projections $X\times X\rightrightarrows X$ exists then $e$ determines a morphism $T\to X$ so that $X$ will become a "usual" monoid in ${\mathbf C}/T$. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Dec 22, 2017 at 17:36
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    $\begingroup$ Assuming classical logic, the map from any a non zero locale to the point is an open surjection, hence a regular epimorphism this implies that any constant map factor through the point and hence this notion is in this case equivalent to the usual notion of monoid. In other category or constructively the notion can be non trivial and interesting though. I never saw it before. $\endgroup$
    – Simon Henry
    Commented Dec 22, 2017 at 17:49
  • $\begingroup$ @SimonHenry Yes, I'm working constructively. Maybe this notion is useful only in this setting (at least for locales it is). $\endgroup$
    – Valery Isaev
    Commented Dec 22, 2017 at 17:59

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I don't believe there are non-trivial examples of this concept. Assume that $e: X \to X$ is a constant map, then $e$ is idempotent. Any category can be embedded fully faithfully into a Cauchy complete category, so we can assume that $e$ splits as $X \xrightarrow{p} T \xrightarrow{i} X$ with $p\circ i=1_T$. I claim that $T$ is a terminal subterminal object. Indeed, let $f,g: A \to T$, then $$\begin{eqnarray} A \xrightarrow f T & = & A \xrightarrow f \left( T \xrightarrow i X \xrightarrow p T\right) \xrightarrow 1 \left(T \xrightarrow i X \xrightarrow p T \right) \\ & = & A \xrightarrow f T \xrightarrow i \left(X \xrightarrow p T \xrightarrow 1 T \xrightarrow i X \right) \xrightarrow p T \\ & = & A \xrightarrow g T \xrightarrow i \left(X \xrightarrow p T \xrightarrow 1 T \xrightarrow i X \right) \xrightarrow p T \\ & = & A \xrightarrow g T \end{eqnarray} $$

Regarding your supposed example of a locale, it is most likely that what you have is either a groupoid in locales or a localic group over some non-trivial base.

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  • $\begingroup$ There might be $A$ without any maps to $X$, so $T$ is only subterminal. There are categories with lots of subterminals but without a terminal. What is true is that $X$ is a monoid in the slice over $T$ (more or less as in my comment). $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Dec 22, 2017 at 17:46
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    $\begingroup$ The empty set is an exemple in the category of sets. If G is a group object in $C/X$ and $X \rightarrow Y $ is a mono then G is a non trivial exemple of the notion in $C/Y$ $\endgroup$
    – Simon Henry
    Commented Dec 22, 2017 at 17:57
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    $\begingroup$ Hum... So what you proved is that in Cauchy complete category those are exactly the monoid objects of slice categories by subterminal objects. Nice ! $\endgroup$
    – Simon Henry
    Commented Dec 22, 2017 at 18:03
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    $\begingroup$ An example: elementary toposes and geometric morphisms. It has image factorizations, so in particular idempotents split. The topos of finite sets, as well as sets below an inaccessible cardinal, are subterminals. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Dec 22, 2017 at 21:35
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    $\begingroup$ @მამუკაჯიბლაძე : My point was just that the inclusion of the category of finite sets or of sets smaller than an inaccessible cardinals are not geometric morphisms because they do not have an adjoint. But they are logical morphisms. I agree that the category of sets is not terminal in the category of logical morphisms. $\endgroup$
    – Simon Henry
    Commented Dec 24, 2017 at 9:16

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