Fields in monoidal categories

Question

We can speak of rings in monoidal categories, including also the non-Cartesian case. What about fields?

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Question 1: Definitions

What are some possible notions of a (skew or commutative) field in a symmetric monoidal category $\mathcal{C}$?

So far, I've found the following:

• Approach #1: Taking the point of view of fields as rings where nonzero elements have inverses, one could start by considering groups of units, as done in the nLab page on topological fields.
• Approach #2: Viewing fields representation-theoretically, we could define a field in $\mathcal{C}$ to be a ring object $k$ such that every $k$-module in $\mathcal{C}$ is (isomorphic to a) free one.
• Approach #3: The notion of an ideal makes sense in any "nice" monoidal category; see Section 4.2 of Martin brandenburg's PhD thesis. One could define a field in $\mathcal{C}$ as a ring object $k$ in $\mathcal{C}$ having only $k$ and $(0)$ as ideals.

How do these approaches compare to each other? What are other possible definitions?

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Question 2: Examples

Finally, what are some examples of fields in monoidal categories? In particular:

• When $\mathcal{C}=\mathsf{Top}$, the first approach above recovers topological fields (i.e. topological rings $k$ which are fields but also whose inverse map $a\mapsto a^{-1}$ defines a continuous map $(-)^{-1}\colon k^{\times}\to k^{\times}$). Do the other approaches recover this continuity condition too?
• For $\mathcal{C}=\mathsf{Sch}$, we have a well-studied notion of a ring scheme, of which a very important example is the ring scheme $\mathbb{W}$ of Witt vectors (for an introduction, see Eric's translation of Grothendieck's Groupes de Barsotti–Tate et Cristaux de Dieudonné). What are some examples of field schemes?
• For $\mathcal{C}=\mathsf{CCoAlg}_{R}$, rings in $\mathcal{C}$ give the notion of a Hopf ring. What are examples of Hopf fields?

Answer 1

In the stable homotopy category, it is standard to define a field spectrum to be a ring spectrum $F$ such that every nonzero homogeneous element of the graded homotopy ring $\pi_*(F)$ is invertible. This ensures that every $F$-module spectrum is isomorphic to $\bigoplus_{i\in I}\Sigma^{d_i}F$ for some indexing set $I$ and some system of degrees $d_i\in\mathbb{Z}$.

For each prime $p$ and integer $n>0$ there is a Morava $K$-theory spectrum $K(p,n)$ with $\pi_*(K(p,n))=(\mathbb{Z}/p)[v_n,v_n^{-1}]$ with $|v_n|=2(p^n-1)$. Additionally, we have the Eilenberg-MacLane spectra $K(p,\infty)=H\mathbb{Z}/p$ and $K(0)=H\mathbb{Q}$. These are all field spectra. It is an important theorem that in a certain sense (not quite the most obvious one) this is a complete list of the prime field spectra. All of these examples are commutative except in the case $p=2$ where instead we have a rule like $ab-ba=v_nQ_n(a)Q_n(b)$ for a certain operation $Q_n$ with $Q_n^2=0$.

Answer 2

Paul Balmer, Henning Krause, and Greg Stevenson have investigated the concept of 'field' in tensor-triangulated geometry (see Definition 1.1).