I was looking an example of determinant as natural transformation. I have tried to prove it, but my question here is regarding the categories in which we are looking it as a functor.

Fix a natural number $n$. Let Comm denotes the category of commutative rings. For every object $R$ in Comm, we have a new ring $M_n(R)$. Let Mon denotes the category of monoids. Then $M_n(R)$ is an object in the Mon for every object $R$ in Comm.

Now $\mathcal{F}:$ Comm $\rightarrow$ Mon, $R\mapsto M_n(R)$.

Let $\mathcal{G}:$ Comm $\rightarrow$ Mon, $R\mapsto R$ (where $R$ in domain is commutative ring, and $R$ in co-domain is monoid w.r.t. multiplication).

It is then proved that determinant is a natural transformation from $\mathcal{F}$ to $\mathcal{G}$.

I have proved this using the Axiometic definition of natural transformation (I mean, beyond Axiometic verification, I have not understood much).

But, my question here is the following. The objects $M_n(R)$ considered above belong to a category of monoids; but at the same time, they are also rings, and so we can consider them as objects in the category Rings of rings. Is there any specific reason to consider $M_n(R)$ as objects in Mon instead of Rings? In other words,

What problem arises if we consider $\mathcal{F}$ and $\mathcal{G}$ as functors from Comm to Rings?

  • $\begingroup$ A book describing this approach is Leinster: Basic Category Theory, page 29 $\endgroup$ – Javier Apr 17 at 12:54

The problem is that $\det$ is only a multiplicative monoid homomorphism, not a ring homomorphism. That is, $\det(A+B)\neq\det(A)+\det(B)$ in general. So if you considered your functors as taking values in $\mathbf{Rings}$, $\det$ would not even be a morphism in that category for any fixed $R$.

  • $\begingroup$ Eric, thanks for the clarification. $\endgroup$ – p Groups Sep 27 '16 at 8:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.