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Here is an idle thought on those “extra” conditions in cohesion, on top of the bare adjunctions.
Given an idempotent comonad $\Box$ on a topos, we may think of $\Box X$ as a projection of “pure $\Box$-moment” out of $X$. Let’s say therefore that the “negative” (or maybe the “complement”) of $\Box$ is the cofiber of its counit
$\overline{\Box} X \coloneqq cofib(\Box X \to X)$i.e. that which remains after contracting away the “$\Box$-moment”.
From this intuition it makes sense to ask whether this concept of negatives is accurate in that there is indeed no $\Box$-moment left in $\overline{\Box}$. Formally this means to ask whether
$\Box \overline{\Box} \simeq \ast \,.$For general $\Box$ there is no reason for this to be the case. There is more hope when there is also a notion of opposite to $\Box$ in the sense of an adjoint modality to it, either to the left or to the right. Then at least the left adjoint may be taken into the cofiber and something may be said.
Consider hence an opposition of the form $\bigcirc \dashv \Box$, Notice that for oppositions like this, both $\bigcirc$ and $\Box$ express “the same moment” but opposite ways of projecting onto it. (Think of the example where $\bigcirc$ and $\Box$ are the opposite moments of integrality in real numbers given by $floor$ and $ceiling$). This motivates to ask that $\overline{\Box}$ contains no $\bigcirc$-moment (which is the same kind of moment as $\Box$-moment, but projected out in a way so that the desired statement has a chance to be analysable.)
That leads to the question: what are sufficient and what are necessary conditions on an adjoint modality $\bigcirc \dashv \Box$ such that
$\bigcirc \overline{\Box} \simeq \ast$?
I am not sure about necessity, but in a 1-topos, one sufficient condition that comes to mind is the condition that:
$\bigcirc \ast \simeq \ast$
$\Box \to \bigcirc$ is epi
Because with the first item then
$\bigcirc \overline{\Box} X = \bigcirc cofib(\Box X \to X) \simeq cofib(\Box X \to \bigcirc X)$and with the second item this is an object that receives an epimorphism from the point, wich forces it to be the point itself, doesn’t it.
What is maybe interesting here is that these conditions are just the extra conditions in “Axiomatic cohesion” on top of the bare adjunctions themselves (the remaining one being that $\bigcirc$ also respects binary products).
This argument of course crucially relies on being in a 1-topos, not in an $(\infty,1)$-topos. On the other hand in an $(\infty,1)$-topos there is another kind of statement which expresses roughly the same intuition, that $\bigcirc$ is complementary to $\overline{\Box}$: namely fracturing says that each stable object is a fibered direct sum of its $\bigcirc$-moment with its $\overline{\Box}$-moment
$X \simeq (\bigcirc X) \underset{\bigcirc \overline{\Box}X}{\oplus} (\overline{\Box} X) \,.$Regarding the example with floor and ceiling, does this make $U(1)$ turn up as cofibre of $\mathbb{Z} \to \mathbb{R}$? I’ve always regarded as using $U(1)$ as the default infinitely deloopable cohesive infinity group as ad-hoc, whereas if it turns up from such axiomatics as you outline for $\overline{\Box}$, and the integers and reals, then this seems more satisfactory.
It also seems more in line with the naïve approach that physicists take when they get a real number, only specified up to some ambiguity, which they call some sort of anomaly, and then they demand this is an integer to get a well-defined object, aka exactly an element of the cofibre of $\mathbb{Z} \to \mathbb{R}$. That one calculates this via an exponential is then a matter of parameterisation/models…
For that simple illustrative example to work (see at adjoint modality – Simple illustrative examples) one needs to think of $\mathbb{Z}$ and $\mathbb{R}$ as categories qua their canonical linear ordering. This is just to illustrate how a $\bigcirc \dashv \Box$ is an opposition, otherwise it falls outside of the discussion above, as these categories are not toposes (they don’t even have terminal objects). Maybe it’s a red herring that you should ignore if it doesn’t seem helpful.
If we do go into a cohesive topos, then to motivate $\mathbb{R}$ we may ask that it represents the cohesion in that $L_{\mathbb{R}}\simeq shape$. Given that, then in practice the problem arises of descending $\mathbb{R}$-valued cocycles along projections of their domain, for instance $n$-truncation of universal Lie integrations $\exp(\mathfrak{g}) \stackrel{\tau_n}{\longrightarrow} \mathbf{B}G$. This in general only works up to quotienting out a subgroup $\Gamma \hookrightarrow \mathbb{R}$ (of periods of the Lie cocycle in the given example)
$\array{ \exp(\mathfrak{g}) &\longrightarrow& \mathbf{B}^{n}\mathbb{R} \\ \downarrow && \downarrow \\ \mathbf{B}G &\longrightarrow& \mathbf{B}^n (\mathbb{R}/\Gamma) }$Now either $\Gamma \hookrightarrow \mathbb{R}$ dense, then it is not really interesting; or it is not dense, then it is the intergers, I’d think.
This is one way to see how $U(1)$ arises in physics, already in prequantum physics. For instance with a symplectic potential 1-form on a phase with with $G$-action, then it is unlikely to descend as an equivariant form, but it is more likely to descend as a $\mathbb{R}/\Gamma$-principal connection, with $\Gamma \hookrightarrow \mathbb{R}$ the discrete subgroup of failures of the naive descent.
In any case, this is not related to determinations of moments and their determinate negation, which I am after here. If you feel like discussing the ontology of $U(1)$ further, then let’s do it in another thread!
Yes, I was thinking as internal categories, but then of course floor etc is not continous, so not an internal functor. I asked anyway, in case there was some way of making sense of the notion.
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