# Is there a technical term for a category that is equipped with a functor into a fixed group?

In the example I have in mind (which is very out-of-the-way, so that giving details of it would only obscure the question), the 'fixed' group $S$ has two elements, which can be identified with $+1$ and $-1$ under ordinary integer multiplication, so the existence of the functor $\sigma: \mathcal{C} \to S$ means (i) every map $f$ in $\mathcal{C}$ is either 'positive' ($\sigma(f) = +1$) or 'negative' ($\sigma(f) = -1$), but not both, (ii) the 'signs' of maps satisfy $\sigma(fg) = \sigma(f)\sigma(g),$ for all $f, g$ composable in $\mathcal{C},$ and (iii) the identity maps in $\mathcal{C}$ are all 'positive'.

I'd like to know if there is a technical term for a category such as $\mathcal{C}$ here, preferably with this particular group $S$, or alternatively with a general group $S$ (although this allows the trivial case where $|S| = 1$).

• It's called a graded category for the general case. You could probably use "supercategory" for the $\mathbb{Z}_2$ case. – Derek Elkins Jul 6 '17 at 17:07
• @DerekElkins Thank you very much! If no-one else posts an answer in the next day or so, please repost your comment as an answer, and I'll accept it. – Calum Gilhooley Jul 6 '17 at 17:49
• Hmm.. If you rather wanted to map to the catgeory of 2 objects and an inverse pair of arrows, it also deals the objects into 2 classes. Such can be called a Morita context between the two designated full subcategories (the preimages of the 2 identity maps). – Berci Jul 6 '17 at 22:22

The category you're mapping into should be called $B \mathbb{Z}_2$. In general, you can call a category $C$ equipped with a functor into another category $D$ a category over $D$, so in this particular case you can refer to categories over $B \mathbb{Z}_2$.
• Could you explain what "$B\mathbb{Z}_2$" means? All I could find using Google was a comment attached to a mathoverflow question saying "$B\mathbb{Z}_2$ can be chosen as a $\mathbb{Z}_2$-module plus some topology", and a Wikipedia article saying "An example of a classifying space is that when $G$ is cyclic of order two; then $BG$ is real projective space of infinite dimension"- both much more advanced than treating a group or monoid as a category with 1 object! – Calum Gilhooley Jul 8 '17 at 0:02
• @Calum: if $G$ is a group (or monoid), $BG$ is the category with one object and morphisms given by $G$. It is really important to distinguish this from $G$ itself, which is a set, not a category. The relation to topology is that there is a functor from categories to topological spaces called taking the geometric realization of the nerve, and it sends $BG$ the category to $BG$ the space. – Qiaochu Yuan Jul 8 '17 at 1:21
• So, writing something like $\sigma: \mathcal{C} \to S$ is not a permissible "abuse of language"? I live and learn! – Calum Gilhooley Jul 8 '17 at 2:19
• @Calum: the temptation is understandable but I think it is ultimately not a good convention, and the $B$ convention is better. – Qiaochu Yuan Jul 8 '17 at 2:25
• @CalumGilhooley It's called delooping. The general idea is probably not very accessible (and I can easily see the nLab page coming off as intimidating), but in this case you can view $B$ as the functor $B : \mathbf{Grp}\to\mathbf{Cat}$ (or into $\mathbf{Grpd}$, the category of groupoids and functors) which implements the usual "a group is a one-object category/groupoid". From this latter perspective, it simply avoids an abuse of notation akin to how it's better to say $DS$ for the discrete category generated by $S$. – Derek Elkins Jul 8 '17 at 23:31