# What's that category?

31 Oct 2016Some vague thoughts about a weird category me and my housemate got to thinking about recently, unfortunately I’m a little too sleepy to write anything more coherent right now, but beeminder demands tribute.

When doing non-abelian group cohomology (and many other things) you end up dealing with the category of pointed sets, that is, sets with a point specified, and where morphisms must map specified points to each other.
This category is actually fairly nice (or at least nicer than plain $\mathrm{Set}$ is anyway) insomuch as it has a zero object (i.e. an object that is both initial and terminal), the one element set, this allows us to make sense of kernels etc. which is a nice thing to be able to talk about.
So why do we get a 0-object here? Well the one element set is already terminal in $\mathrm{Set}$ so we get terminalness for free, as for *why* it is initial it’s revealing to describe the category of pointed sets in a different way:
We can equivalently describe it as the coslice category $\{* \} \downarrow \mathrm{Set}$, where the objects are morphisms in $\mathrm{Set}$ from the one element set (giving you the specified point) and the new morphisms are commuting triangles in $\mathrm{Set}$.
When we do this we of course get the object we cosliced at (or more specifically its identity morphism) as an initial object, and as we already had a unique map from any object to this object we get that its identity map to itself is terminal also.

This got me thinking about $\mathrm{CRing}$, which infamously doesn’t have kernels, so what if we follow the recipe above or at least it’s dual and consider the slice category over the initial object $\mathbf{Z}$. By the dual of the above this should now have a 0-object (the identity map on $\mathbf{Z}$) and so we could form kernels, indeed the kernel of a morphism $A \to B$ (where both $A$ and $B$ have an associated map to $\mathbf{Z}$) should be the preimage of $\mathbf{Z}$ I suppose, and this is probably a subring?

Before we get to that though be should ask what does this category $\mathrm{CRing}\downarrow \mathbf{Z}$ even look like?! It’s not absolutely trivial as far as I can tell as we get maps from polynomial rings into $\mathbf{Z}$ from evaluating at integer vectors. We can also add in nilpotents and other stuff that just ends up mapping to zero but is still fine. However on the other hand we immediately rule out positive characteristic rings, and anything which inverts an element of $\mathbf{Z}$.

I really have no conclusions about what this category of commutative rings with a map to $\mathbf{Z}$ actually is, but it is quite fun to play with.