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Let $C$ be a category with coproducts and terminal object $1$ such that every object in $C$ is isomorphic to a coproduct of $1$ with itself (indexed over some set). Is there a special name for such a category?

EDIT: Interested specifically in the case in which $C$ is a topos.

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  • $\begingroup$ A subcategory of Sets? $\endgroup$
    – egreg
    Oct 27, 2016 at 23:22
  • $\begingroup$ @egreg Wouldn't it rather be something like a "quotient" of Sets instead, via $J\mapsto\bigvee_J 1$ on objects, and with some natural definition on functions (which I don't want to write here)? $\endgroup$ Oct 27, 2016 at 23:25
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    $\begingroup$ @LuizCordeiro: There is a canonical functor from $\mathbf{Sets}$ to any such category which is essentially surjective on objects, but it need not be a full functor. For instance, the full subcategory of $\mathbf{Sets}^2$ consisting of objects of the form $(X,X)$ is one such category, but the image of your "quotient" functor contains only the morphisms that are the same on both coordinates. $\endgroup$ Oct 27, 2016 at 23:50
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    $\begingroup$ @egreg is right: The functor from $C$ to the category of sets which maps an object $X$ to the set $\mathrm{Hom}_C(1,X)$ of global elements of $X$ is fully faithful. (@egreg, are you Greg Egan, my favourite science fiction author?) $\endgroup$ Jun 28, 2018 at 2:28
  • $\begingroup$ @IngoBlechschmidt Sorry, no. 😊 $\endgroup$
    – egreg
    Jun 28, 2018 at 7:36

1 Answer 1

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These are precisely the cocartesian categories $\mathbf{C}$ for which the (singleton set containing the) terminal object $\{ 1 \}$ is a colimit-dense generator. See, for instance, §III.7 of Adámek–Tholen's Total Categories with generators. I do not believe there is a more specific name; examples seem sparse.

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