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The Baez–Dolan microcosm principle is stated in the nLab as follows.

Microcosm principle: Certain algebraic structures can be defined in any category equipped with a categorified version of the same structure.

Recently I noticed that, while we can define rings in monoidal categories (see here for examples) via a somewhat contrived construction, there's a much simpler notion of a semiring in a semiring category, detailed in the nLab, with examples.

Question. Has this notion been defined/studied before in the literature?

Edit. Yes, it partially appeared in Definition 5.1 in

Morten Brun, Witt Vectors and Equivariant Ring Spectra, 2006. Proceedings of the London Mathematical Society, Volume 94, pp. 351–385. (arXiv:math/0411567, doi:10.1112/plms/pdl010.)

but somewhat quickly. Are there other references?

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    $\begingroup$ All your examples have $\oplus= $ coproduct, so that the additive monoid structure trivializes. Do you have examples in mind where it doesn't ? $\endgroup$
    – Maxime Ramzi
    Commented Sep 2, 2021 at 7:23
  • $\begingroup$ There's also a page on 2-rigs -- ncatlab.org/nlab/show/2-rig -- which presents a range of variants. $\endgroup$
    – David Corfield
    Commented Sep 2, 2021 at 8:03
  • $\begingroup$ @MaximeRamzi Currently, no :/ (besides trivial ones like $0$ with the trivial morphisms in $\mathbb{F}$, $\mathbb{S}$, or $R_{\mathsf{disc}}$ for $R$ a ring). In fact, I think I don't know many semiring categories with $\oplus\neq$ coproduct at all! I've asked about this in a separate question here, after failing to come up with any useful examples... $\endgroup$
    – Emily
    Commented Sep 3, 2021 at 7:08
  • $\begingroup$ This notion appears also here (and in more generality!): ncatlab.org/nlab/show/colax-distributive+rig+category. $\endgroup$
    – Emily
    Commented Oct 8, 2021 at 17:42

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