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I have a group with a category structure (i.e. category whose objects form a group), such that the left multiplication with any fixed element is a category automorphism. The same is true for right multiplication. Is there a well-known name for that kind of groups? Any references?

If the category can be interpreted as an ordered set, the group is called ordered group. So what is the name of the generalisation to categories?

Note: For those who consider groups as one-object categories: In this setting I'd like to know how a 2-category is called that is a group.

Edit: Clarify group with category structure.

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    $\begingroup$ What does "a group with a category structure" mean? $\endgroup$ Jul 18 '19 at 19:52
  • $\begingroup$ It sounds like you're taking about something like $G-Set$ for some group $G$. For an arbitrary category $C$, the general setting would be functors $G \to C$ where $G$ is construed as a one-object category whose morphisms are all isos. $\endgroup$ Jul 18 '19 at 21:40
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    $\begingroup$ I think groupoidal category (the group analogue of monoidal category) is sometimes used, although the "systematic" name for the closest thing that I think exists would probably be something like "group object in the category of categories" (although your assumptions are not quite enough for that) $\endgroup$ Jul 18 '19 at 21:51
  • $\begingroup$ A "group object in the category of categories" is often called a $cat^1$-group or a strict $2$-group (and it's equivalent to a crossed module). $\endgroup$
    – Arnaud D.
    Jul 19 '19 at 6:02
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    $\begingroup$ In the nlab, this sort of structure is also referred to as "groupal categories" or "groupal groupoids" $\endgroup$ Jul 19 '19 at 8:42
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The correct definition requires some care around what it means for such a thing to be associative. It turns out that associativity requires extra data: you are looking for a monoidal category, which involves extra maps called associators, and you also want the monoidal structure to be grouplike, which means that every object is invertible. So I would call such a thing a grouplike monoidal category. It's somewhat awkward terminology but all of the other options I know are similarly awkward.

Every monoidal category gives rise to a grouplike monoidal category given by taking the subcategory of invertible objects. A simple example is the category of $R$-modules for $R$ a commutative ring; taking invertible objects gives the grouplike monoidal category of invertible $R$-modules, or line bundles over $\text{Spec } R$.

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  • $\begingroup$ What is the benefit over “groupal category” as suggested by nlab via Thibaut Benjamin in the comments? Regarding your secound statement: You are right: The set of invertible elements (units) of a monoid always forms a group, sometimes the trivial group. I must admit, as my background is neither category nor ring theory, I fail to recognize the simplicity in your examples. $\endgroup$
    – Keinstein
    Sep 13 '19 at 18:57
  • $\begingroup$ "Grouplike" is consistent with terminology from homotopy theory (where people talk about e.g. grouplike $E_n$-algebras); other than that it's a matter of personal preference for me, I find it less awkward than "groupal." $\endgroup$ Sep 13 '19 at 19:00

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