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I don’t understand the intent of the definition of infinitesimal cohesive (infinity,1)-topos. Is it a cohesive one equipped with equivalences $ʃ \simeq \flat \simeq \sharp$? Should those equivalences be coherent with the structure of the modalities in some way? Is there an ambijunction going on?
Hm, I am thinking of there just existing an equivalence (of functors, hence a natural equivalence). Is there some trouble lurking which I am overlooking?
Well, as a category theorist, the phrase “there exists an equivalence” always makes me suspicious. E.g. someone may say that in a monoidal category “there exists an isomorphism $x\otimes (y\otimes z)\cong (x\otimes y)\otimes z$”, but they really should mean that there is a specified such isomorphism satisfying coherence conditions.
Okay, sure, and maybe I am missing something. But we are speaking of adjoints here, which are unique up to a contractible space of choices. I’d think that makes it work.
Sure. But just because (say) right adjoints of a fixed functor are unique up to unique isomorphism doesn’t mean that there is only one possible isomorphism between the underlying functors. Even if you fix the underlying functor there may be more than one way of making it into a right adjoint. And the situation we are in is even worse: we have to compare left and right adjoints. In the situation where we have $\Gamma \dashv \Delta \dashv \Gamma$, we can canonically define a natural transformation like $id \Rightarrow \Delta \Gamma \Rightarrow id$ (as Lawvere does), which is not possible if we only have $\Gamma \dashv \Delta \dashv \nabla$ with an unspecified isomorphism $\Gamma \cong \nabla$.
Hm, “even worse”, “not possible”? Can you point out an actual technical problem?
To me it seems that pretending that we can say that the left and right adjoint are actually equal would be a problematic step.
But please convince me otherwise. What goes actually wrong with the definition that I have given?
Well, what do you want to achieve? Lawvere isn’t very explicit, but I’m guessing based on Proposition 1 in [Axiomatic cohesion] that there’s an unstated coherence condition that the composite $\Delta \Gamma \Rightarrow id \Rightarrow \Delta \Gamma$ is not merely an isomorphism but the identity (which is certainly the case for the quality type $\operatorname{Ho} Kan$), so that the other composite $id \Rightarrow \Delta \Gamma \Rightarrow id$ is idempotent.
I’d rather not say that anything is an identity.
There is nothing much to achieve here, it’s just a piece of terminology. In a cohesive topos in which there is an equivalences $\Pi \simeq \flat$ every object behaves like an infinitesimally thickened point, and that deserves a name. The examples show that this shows up naturally in useful contexts, so we want to be able to speak about it.
Also, I wouldn’t think that anything Lawvere does in his article relies on being evil and pin-pointing identities.
Then how do you propose to prove Proposition 1, that a quality type has a central idempotent? I don’t see any other candidates for a central idempotent lying around, and it’s not even obvious whether $id \Rightarrow \Delta \Gamma \Rightarrow id$ is an idempotent without assuming something extra. And I think you will agree with me that this should, in all geometric examples, be an idempotent.
Okay, I think I’ve figured it out. Thanks Zhen for pointing to that idempotent that Lawvere wants.
In general, while to say that a functor has a left adjoint, or has a right adjoint, is a mere property, to say that it has a ambidextrous adjoint (one functor that is both left and right adjoint) is a structure — e.g. the structure of a chosen isomorphism between its left and right adjoints. However, if a functor $p^*$ is fully faithful, then there is a stronger sort of “ambidextrous reflection” which is a mere property, namely to ask that the canonical map $p_* \to p_!$ from its right to its left adjoint is an isomorphism. I think this must be what Lawvere has in mind, because it does suffice to make that canonical map $Id \to p^* p_! \leftarrow p^* p_* \to Id$ an idempotent. In other words, an infinitesimally cohesive topos is one in which “pieces are points”.
Speaking internally, this means we have a full subcategory of discrete objects which is both reflective with reflector $ʃ$ and coreflective with coreflector $\flat$, plus the induced map $\flat A \to A \to ʃ A$ is an isomorphism for any $A$. The central idempotent is then the map $A \to ʃA \cong \flat A \to A$. Among other things, we can then conclude that $ʃ$ is left exact, and hence can also be called $\sharp$.
Does that seem right?
Have to rush, but just quickly on the English language description: I wouldn’t say “pieces are points” since the piece is the piece with its internal structure. I’d stick with “there is precisely one point per piece” or “precisely one point per cohesive neighbourhood”.
Okay, I kept thinking that there being any equivalence and the points-to-pieces transform being an equivalence is equivalent. But now I keep having trouble showing this, and probably it’s not true.
Thanks, then I get now what the issue is.
I had a think about it last night. I think the ordinary analogue is true when:
I haven’t checked all the details, but the main idea is that $\mathcal{F}$ should be equivalent to $Fam (\mathcal{C})$, where $\mathcal{C}$ is the full subcategory of objects $A$ in $\mathcal{F}$ such that $\Gamma A$ is a singleton. Observe that $\pi_0$ is the unique (up to unique isomorphism) coproduct-preserving functor that sends objects in $\mathcal{C}$ to singletons. Now, if $\Gamma \cong \pi_0$ (not necessarily canonically), then the canonical $\Gamma \Rightarrow \pi_0$ is an isomorphism for objects in $\mathcal{C}$, hence, for all objects, since $\Gamma$ and $\pi_0$ preserve coproducts.
Notice that $\Delta 1$ is an initial object in $\mathcal{C}$, while $\nabla 1$ is a terminal object in $\mathcal{C}$. I expect that $\mathcal{F}$ is a quality type over $Set$ if and only if $\mathcal{C}$ has a zero object. I also suspect it can be generalised if we assume that the $\mathcal{S}$-indexed category induced by $\Delta : \mathcal{S} \to \mathcal{F}$ is a stack for a suitable coverage.
It certainly could be one of those places where having a noncanonical isomorphism implies that the canonical map is an isomorphism. But regardless, the definition should probably be that the canonical map is an iso.
Urs #11 you’re right. Maybe “pieces are halos” or something (-:
Or, “cohesive pieces are uniquely pointed”? Where ’unique’ is appropriately defined.
added pointer to Johnstone 96 on “quintessential localization”
(thanks to Thomas Holder for highlighting the reference)
I added to the Idea
Infinitesimal cohesion may also be defined relative to any (∞,1)-topos.
I’ve been trying to get a conversation going at the $n$-Café, here, about related matters.
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