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Martti Karvonen (Feb 05 2023 at 15:25):Inspired the [[microcosm principle]], I'd like to know what sort of algebraic structures fit natively inside rig categories, just like (commutative) monoids live inside (symmetric) monoidal categories. So is there a known notion of a rig object inside a rig category? It looks like one might want to use a different monoidal structures for multiplication addition, which sounds funny to me - do such things actually show up somwhere?
John Baez (Feb 05 2023 at 17:52):Great question! One thing worth looking at is the usual concept of ring defined inside the rig category AbGrp. Say we have a ring thought of as an object in AbGrp. Then addition gives a morphism
while multiplication gives a morphism
John Baez (Feb 05 2023 at 17:53):So two different monoidal structures are getting used here, for + and for , in a microcosmic way.
Nathanael Arkor (Feb 05 2023 at 18:13):One natural definition is given by Martin Brandenburg here.
Martti Karvonen (Feb 05 2023 at 18:59):Nice! I was too stuck with the idea that usual rings are internal to Set, instead of being internal to Ab.
Martti Karvonen (Feb 05 2023 at 19:22):I guess these would also coincide with some kind of lax morphisms from the terminal rig cat to the rig cat in question?
John Baez (Feb 05 2023 at 20:55):I don't know... lax rig category maps? Maybe.
John Baez (Feb 05 2023 at 20:57):Note that what I said is a bit odd because while you can treat a ring as a rig object in the rig category you can also more simply treat it as a monoid object in the monoidal category - you then get the addition, and distributivity of multiplication over addition, "for free".
Mike Shulman (Feb 06 2023 at 04:58):That's presumably because is the cocartesian monoidal structure on , so every object is a commutative monoid for it in a unique way.
Mike Shulman (Feb 06 2023 at 04:59):The definition of "semiring object in a bimonoidal category" is written out at [[ring object]]. But I observe that all the examples given there also have this property, that the "additive" monoidal structure is cocartesian and hence is not doing any work.
Mike Shulman (Feb 06 2023 at 05:00):(A different way to internalize a ring object is in a [[duoidal category]].)
John Baez (Feb 06 2023 at 06:04):Mike Shulman said:
The definition of "semiring object in a bimonoidal category" is written out at [[ring object]]. But I observe that all the examples given there also have this property, that the "additive" monoidal structure is cocartesian and hence is not doing any work.
The rig categories I've been studying intensively have this property: they are Cauchy complete symmetric monoidal -linear categories, where the additive monoidal structure is cocartesian. It's "doing work" in some sense, but it's not an extra structure, just a property.
Mike Shulman (Feb 06 2023 at 08:42):A monoid structure for a cocartesian monoidal structure isn't even an extra property, it's whatever comes below that: it always exists uniquely. That's what I meant: when the additive monoidal structure is cocartesian, a "rig object" is no different than a monoid object for the multiplicative monoidal structure, so the additive strurcture may as well not be there for purposes of defining rig objects.
Matteo Capucci (he/him) (Feb 06 2023 at 12:12):It's (-1)-stuff, cool! (edit: it's (-2)-stuff)
John Baez (Feb 06 2023 at 16:44):Okay, I meant that cocartesianness on the category is an extra property. But yes: for the poor hapless object of such a category, having a monoid structure with respect to + isn't even an extra property, it's an extra automatically true property.
John Baez (Feb 06 2023 at 16:45):Here "extra" is being used in a somewhat sarcastic way.