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Let $\mathbf{H}$ be a topos and $\mathbb{G}$ a monoid object in $\mathbf{H}$. Then the slice topos $\mathbf{H}_{/\mathbb{G}}$ carries in addition to its canonical cartesian monoidal structure $\times_{\mathbb{G}}$ a non-cartesian monoidal structure $\otimes_{\mathbb{G}}$ given on objects as
$\left[ \array{ X_1 \\ \downarrow^{\mathrlap{\chi_1}} \\ \mathbb{G} } \right] \otimes_{\mathbb{G}} \left[ \array{ X_2 \\ \downarrow^{\mathrlap{\chi_2}} \\ \mathbb{G} } \right] \;\; \coloneqq \;\; \left[ \array{ X_1 \times X_2 \\ \downarrow^{\mathrlap{(\chi_1, \chi_2)}} \\ \mathbb{G} \times \mathbb{G} \\ \downarrow^{\mathrlap{\cdot}} \\ \mathbb{G} } \right] \,.$One might call $(\mathbf{H}_{/\mathbb{G}}, \otimes_{\mathbb{G}})$ a “monoidal topos”.
(There is also the proposal around to call a topos with a suitable extra tensor product structure a “quantum topos”, though that term has its dangers. Moreover, judging from applications, at least those monoidal toposes of the above slice form would much rather be called prequantum toposes, if at all.)
While this is easy enough, the question is if this is a kind of structure that deserves to be considered a bit more in its own right.
For instance one might ask which properties of $(\mathbf{H}_{/\mathbb{G}}, \otimes_{\mathbb{G}})$ one might want to abstract and look for non-slice toposes equipped with an extra tensor product that share these.
For instance the category of dendroidal sets equipped with its Boardman-Vogt tensor product is also a “monoidal topos”, and it is not of the above slice form. Is there a useful way to abstractly distinguish it from slice monoidal toposes? Or maybe should it be regarded as on par with them, for some good class of purposes?
So this is a super-vague question: has anyone had thoughts anong these lines? Does that ring a bell with anyone? Does this make you think of something else or other that might be noteworthy?
For entertainment: googling “monoidal topos” yields essentially two hits.
One is an article by some Majkic who has a category modelling databases which satisfies axioms like those of an elementary topos but with exponential objects replaced by internal homs with respect to some non-cartesian monoidal structure.
The other is a page on Jim Dolan’s blog, here. Not exactly sure what it is referring to, there. Is it a topos of sheaves over a monoidal category and equipped with the Day convolution tensor product?
I have created monoidal topos, just for the heck of it.
Okay, now I get it: Jim Dolan is talking about a (semi?)monoidal topos of “toric quasicoherent sheaves” as indicated at
Another important monoidal topos is the classifying topos of objects. It’s essentially the category of finitary endofunctors of Set, and a monoid therein is a Lawvere theory. Richard Garner recently wrote about enrichment over this monoidal category here.
Thanks, Mike! I have added that as an example to monoidal topos now.
Is this line of enquiry to do with gradings? What could be done for sphere spectrum grading in a higher setting?
[Oh, I see know, it’s to do with motivic quantization.]
Yes, I am working on that entry motivic quantization, wanted to announce it a little later when it has taken shape a bit more. But maybe it’s good to have it out in the open now.
So, some colleagues of mine over coffee are wondering about “quantum toposes” or “tensor toposes”, namely monoidal toposes and what they might achieve for quantum theory. On the one hand monoidal Bohr toposes are thought to have application to quantum mechanics of composite systems, on the other hand the topos of dendroidal sets with its Boardman-Vogt tensor product is floating around and happens to also be a monoidal topos,too. This led to the question whether these are usefully thought of jointly in some general theory of monoidal toposes.
Then it struck me that in my work on “motivic quantization” with Joost Nuiten, some monoidal $\infty$-toposes play a pivotal role, as precisely the toposes “of local action functionals”. In a nice and precise way they exactly interpolate between the “nonlinear” toposes of moduli stacks of field configurations, and the “linear” categories of modules of quantum states.
When I mentioned this I was asked if from this class of examples I can see some general axioms on monoidal toposes that would be worthwhile to consider. And for the time being I cannot. So I thought I’d sort of forward this question to here and see if anyone has any thoughts on this.
Added a kind of class of examples to this entry.
Thanks, Todd!!
A thought: I suppose the monoidal structure on $\mathbf{H}_{/\mathbb{G}}$ for $\mathbb{G}$ a group object is closed monoidal. For $f \colon X \to \mathbb{G}$ and $h \colon Y \to \mathbb{G}$ two objects, the internal hom is over $g \in \mathbb{G}$ the space of maps $\phi \colon X \to Y$ such that $(\phi^\ast h) \cdot f^{-1} = g$.
Need to think about how to say this more formally…
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