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Began stub for Tambara functor. Neil Strickland’s, Tambara Functors, arXiv:1205.2516 seems to be a good reference.
Seems like it’s very much to do with pullpush through polynomial functors, if you look around p. 23.
I would try to say what the idea is, but have to dash.
I extracted some further statements by Strickland.
p. 24 of his article has $X \overset{p}{\leftarrow} A \overset{q}{\to} B \overset{r}{\to} Y$ as in polynomial functor which produces the same map composed of pullback of $p$, dependent product for $q$, dependent sum for $r$.
That’s a very interesting catch you made there. I’ll look into this, thanks for pointing this out.
And it all links to Witt vectors, and so then back to arithmetic, as noted here.
According to Joachim Kock Notes on polynomial functors, p. 192, Tambara first looked at a category which is the Lawvere theory for commutative semirings. So that fits with Strickland:
One way to think about the definition of Mackey functors (for a finite group G) is as follows. Take the Lawvere theory A for (commutative) semigroups, categorify it, make it G-equivariant, decategorify, and then take the category of models for the resulting (multisorted) theory. This category is just the category of semigroup-valued Mackey functors, so it contains the more usual category of group-valued Mackey functors.
…
The category of Tambara functors can be defined along the same lines as suggested above for Mackey functors. One simply starts with the Lawvere theory U for semirings instead of the theory A for semigroups.
I added some more bits of pieces. Still not really sure what they’re for.
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