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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeOct 24th 2013

    I don’t understand the intent of the definition of infinitesimal cohesive (infinity,1)-topos. Is it a cohesive one equipped with equivalences ʃʃ \simeq \flat \simeq \sharp? Should those equivalences be coherent with the structure of the modalities in some way? Is there an ambijunction going on?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 24th 2013
    • (edited Oct 24th 2013)

    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?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeOct 24th 2013

    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(yz)(xy)zx\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.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 24th 2013
    • (edited Oct 24th 2013)

    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.

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeOct 24th 2013

    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 ΓΔΓ\Gamma \dashv \Delta \dashv \Gamma, we can canonically define a natural transformation like idΔΓidid \Rightarrow \Delta \Gamma \Rightarrow id (as Lawvere does), which is not possible if we only have ΓΔ\Gamma \dashv \Delta \dashv \nabla with an unspecified isomorphism Γ\Gamma \cong \nabla.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 25th 2013
    • (edited Oct 25th 2013)

    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?

    • CommentRowNumber7.
    • CommentAuthorZhen Lin
    • CommentTimeOct 25th 2013
    • (edited Oct 25th 2013)

    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 ΔΓidΔΓ\Delta \Gamma \Rightarrow id \Rightarrow \Delta \Gamma is not merely an isomorphism but the identity (which is certainly the case for the quality type HoKan\operatorname{Ho} Kan), so that the other composite idΔΓidid \Rightarrow \Delta \Gamma \Rightarrow id is idempotent.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 25th 2013

    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 Π\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.

    • CommentRowNumber9.
    • CommentAuthorZhen Lin
    • CommentTimeOct 25th 2013

    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 idΔΓidid \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.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeOct 25th 2013

    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 *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 *p !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 Idp *p !p *p *IdId \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 ʃʃ and coreflective with coreflector \flat, plus the induced map AAʃA\flat A \to A \to ʃ A is an isomorphism for any AA. The central idempotent is then the map AʃAAAA \to ʃA \cong \flat A \to A. Among other things, we can then conclude that ʃʃ is left exact, and hence can also be called \sharp.

    Does that seem right?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 25th 2013

    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”.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 25th 2013

    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.

    • CommentRowNumber13.
    • CommentAuthorZhen Lin
    • CommentTimeOct 25th 2013
    • (edited Oct 25th 2013)

    I had a think about it last night. I think the ordinary analogue is true when:

    • the base is SetSet, and
    • the upstairs category, say \mathcal{F}, is infinitary extensive.

    I haven’t checked all the details, but the main idea is that \mathcal{F} should be equivalent to Fam(𝒞)Fam (\mathcal{C}), where 𝒞\mathcal{C} is the full subcategory of objects AA in \mathcal{F} such that ΓA\Gamma A is a singleton. Observe that π 0\pi_0 is the unique (up to unique isomorphism) coproduct-preserving functor that sends objects in 𝒞\mathcal{C} to singletons. Now, if Γπ 0\Gamma \cong \pi_0 (not necessarily canonically), then the canonical Γπ 0\Gamma \Rightarrow \pi_0 is an isomorphism for objects in 𝒞\mathcal{C}, hence, for all objects, since Γ\Gamma and π 0\pi_0 preserve coproducts.

    Notice that Δ1\Delta 1 is an initial object in 𝒞\mathcal{C}, while 1\nabla 1 is a terminal object in 𝒞\mathcal{C}. I expect that \mathcal{F} is a quality type over SetSet if and only if 𝒞\mathcal{C} has a zero object. I also suspect it can be generalised if we assume that the 𝒮\mathcal{S}-indexed category induced by Δ:𝒮\Delta : \mathcal{S} \to \mathcal{F} is a stack for a suitable coverage.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeOct 26th 2013

    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 (-:

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 28th 2013

    Or, “cohesive pieces are uniquely pointed”? Where ’unique’ is appropriately defined.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2014

    added pointer to Johnstone 96 on “quintessential localization”

    (thanks to Thomas Holder for highlighting the reference)

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 13th 2021

    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 nn-Café, here, about related matters.

    diff, v17, current

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