In category theory, is there a name/notion for categories where the morphisms have a well-defined parity? Formally, this should mean that there is a non-constant functor from the category to the one-object group $B\Bbb Z_2$.
Alternatively, is there a way to reframe such an idea in a sensible way? Perhaps they decompose naturally into some sort of product?
Here are some examples that come to mind:
- The one-object symmetric groups, where parity is the sign of the permutation
- The category of vector spaces (over some field) restricted to invertible maps, where parity is the sign of the determinant.
- My own motivating example came about when thinking about antilinear maps, i.e. additive maps of complex vector space such that $f(\alpha v)=\bar\alpha f(v)$. One approach that I thought of to put these in a framework was to extend the category of complex vector spaces to include as morphisms both linear and antilinear maps. There, parity is of course given by whether the map is linear or antilinear.
- A monoid (as a one-object category) of even and odd functions under multiplication or composition.
Note: I'm realising now that several of these examples have an ambiguity when it comes to the zero-maps: Are they even or odd? I don't think it matters much, but my gut-feeling tells me that we might as well declare such cases as even.
This answer will essentially be a summary of the comments, although with a few new references to round things off.
A locally graded category is a notion of category in which the morphisms are graded by the objects of a monoidal category $(\mathcal V, \otimes, I)$, such that the grade of a composite $g \circ f$ is the tensor of the grade of $f$ and the grade of $g$, and identity morphisms have grade $I$. In particular, we can grade by a monoid $V$, viewed as a discrete monoidal category. Taking the monoid $\mathbb Z_2$ gives the notion of category whose morphisms have parity you are describing.
There are various different perspectives on locally graded categories. For instance, we can view a locally $\mathcal V$-graded category as a category enriched in the presheaf category $[\mathcal V^\circ, \mathbf{Set}]$ whose monoidal structure is given by Day convolution. When $V$ is simply a monoid, we can view a locally $V$-graded category as a category together with a functor to the delooping of $V$, i.e. the one-object category whose morphisms are elements of $V$ and whose composition is given by the monoid structure. (This is an example of a "graded category" in the terminology of the Wikipedia article, but this terminology does not appear to be common, nor does it appear to be the meaning of "graded category" in the paper that the Wikipedia article cites.)
One further perspective on the specific setting you're interested in is given by Shulman's 2-categories with contravariance, where we take the 2-category to be locally discrete (i.e. given by a 2-category with only identity 2-cells).