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Atrium > Mathematics, Physics & Philosophy: Differential bundles and fibrations for tangent categories

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 12th 2016

    Another paper from the Cockett-Blute-Cruttwell approach to differential category theory:

    • J.R.B. Cockett, G.S.H. Cruttwell, Differential bundles and fibrations for tangent categories, arXiv:1606.08379

    Papers from this approach crop up from time to time on nLab, e.g., at tangent bundle and Kähler differential, but there must be something more to say from the cohesive HoTT POV.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 12th 2016
    • (edited Oct 12th 2016)

    In tems of SDG what they axiomatize is essentially the existence of the functor T=[𝔻 1,−]T = [\mathbb{D}^1, -] acting on microlinear spaces.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 13th 2016

    So although SDG is taken up within smooth cohesive HoTT, do you find no need for microlinear spaces? A quick check just showed 0 appearances of ’microlinear’ in dcct.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 13th 2016
    • (edited Oct 13th 2016)

    One needs the microlinearity condition in order to find additive structure on the fibers of a tangent bundle. But there is a better way to achieve this more generally: instead of the bundle of tangent vectors, consider the infinitesimal disk bundle. This is like the tangent bundle, but remembering that each tangent vector is really to be thought of as an infinitesimal path. Accordingly, there is automatically linear structure on the infinitesimal disk bundle.

    You saw this put to use in Felix Wellen and my talks in Bristol. Instead of formalizing the tangent bundle in Lawvere style as X 𝔻X^{\mathbb{D}}, we formalized the infinitesimal disk bundle as the base change monad operation along the projections from the infinitesimal neighbourhood of the diagonal.

    This is more expressive, as this formalization ends up knowing more about how tangents behave. For instance it also knows everything about PDEs.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 13th 2016

    Right. So I wonder if their other applications would be similarly treatable:

    new examples of tangent categories arising from computer science and combinatorics. In computer science, the resource λ-calculus [Boudol et. al. 1999] and the differential λ-calculus [Ehrhard and Regnier 2003] were developed in parallel. They were, eventually, unified [Manzonetto 2012] as being calculii with their semantics in Cartesian differential categories [Blute et al. 2008]. In combinatorics the differential of a combinatorial species [Bergeron et al. 1997] is also an important tool. This idea was developed more abstractly into a differential of polynomial functors which were then connected to the differential of datatypes (see [Gambino and Kock 2013], [Abbot 2003], and [Abbott et al. 2003]). These provide examples of differentiation in which there is a notion of differentiation but in which negation has no natural meaning. An important aspect of this paper is to spell out in detail the connection between Cartesian differential categories, which provide a unifying framework for the settings above, and tangent categories.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeOct 13th 2016

    Do the infinitesimal disk bundles also yield the abstract structure of “tangent category”?

    How does their additive structure arise synthetically?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 14th 2016
    • (edited Oct 14th 2016)

    The first thing that the axioms of differential cohesion know about the infinitesimal disks 𝔻\mathbb{D} is that they are co-modal for the reduction modality ℜ\Re (I’ll include some hyperlinks, not because you might need them, but just for possible bystandard who might be wondering), i.e.

    ℜ(𝔻)≃*. \Re(\mathbb{D}) \simeq \ast \,.

    This expresses the existence of a 0-element, namely the co-unit 0:*=ℜ(𝔻)→𝔻0 : \ast = \Re(\mathbb{D}) \to \mathbb{D}.

    Next, the axioms know about the additive structure of infinitesimal disks by the product-operation in the infinitesimal disk bundle functor:

    For any Σ\Sigma, write η ΣΣ⟶ℑΣ\eta_\Sigma \;\; \Sigma \longrightarrow \Im \Sigma for the unit of the infinitesimal shape modality. Then the base change monad

    T Σ≔(η Σ) *∘(η Σ) !:H /Σ⟶H /Σ T_\Sigma \coloneqq (\eta_\Sigma)^\ast \circ (\eta_\Sigma)_! \;\colon\; \mathbf{H}_{/\Sigma} \longrightarrow \mathbf{H}_{/\Sigma}

    constructs the infinitesimal disk bundles. Applied to Σ\Sigma itself, this is the infinitesimal version of the tangent bundle of Σ\Sigma, or in fact of its order-kk tangent bundle, if ℑ\Im is interpreted as contracting away order-kk infinitesimal neighbourhoods. For k=∞k = \infty this is formal neighbourhoods.

    (The axioms of differential cohesion do not know which precise value kk has. What may be encoded in the axioms is a sequence of ℑ\Im-operations for different kk of increasing value.)

    Now the product operation in this monad T ΣT_\Sigma, that is a map from infinitesimal disks in infinitesimal disks back to infinitesimal disks

    T ΣT Σ⟶T Σ T_\Sigma T_\Sigma \longrightarrow T_\Sigma

    and in the standard models and for k=1k = 1, this comes out as being addition of tangent vectors.

    I haven’t thought about whether this makes T ΣT_\Sigma satisfy the Blute-Cockett-Cruttwell axioms. Morally their axioms certainly express this kind of situation. But maybe some fine-tuning would be necessary to have a precise comparison.

    • CommentRowNumber8.
    • CommentAuthorMichael_Bachtold
    • CommentTimeOct 14th 2016
    • (edited Oct 14th 2016)

    This sounds really cool, Urs. Could you tell me where I can find this

    Now the product operation in this monad T ΣT_\Sigma, that is a map from infinitesimal disks in infinitesimal disks back to infinitesimal disks

    T ΣT Σ⟶T Σ T_\Sigma T_\Sigma \longrightarrow T_\Sigma

    and in the standard models and for k=1k = 1, this comes out as being addition of tangent vectors.

    worked out in more detail? By analogy with the jet comonad I would have expected that arrow to be of type

    T Σ 1T Σ 1⟶T Σ 2 T^1_\Sigma T^1_\Sigma \longrightarrow T^2_\Sigma

    with the codomain being second order infinitesimal nbhds. But I might be interpreting these disk bundels the wrong way (currently I think of them as the infinitesimal nbhd of the diagonal, as Kock does in his book Synthetic geometry of manifolds.)

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeOct 14th 2016

    I would also like to see more detail. Also, I would have expected addition of tangent vectors to have type T ΣΣ× ΣT ΣΣ→T ΣΣT_\Sigma\Sigma \times_{\Sigma} T_\Sigma\Sigma \to T_\Sigma \Sigma; can we extract that from T ΣT Σ→T ΣT_\Sigma T_\Sigma \to T_\Sigma?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2016
    • (edited Oct 15th 2016)

    Details are in Synthetic variational calculus (schreiber). See example 3.16.

    (This is a joint writeup, in progress, with Igor Khavkine. Dave Carchedi is also involved in the project. But all errors in the file behind the above link are mine.)

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 15th 2016

    Typo poposition in proof there.

    Can we see things intuitively? If the infinitesimal nhbd are pairs of near points, won’t the infinitesimal nhbd of that be pairs of pairs of such points which are near? The first of each pair could be some global point. Then there should be an operation provided by the monad, even when we’re not dealing with some nice patch of real space.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2016
    • (edited Oct 15th 2016)

    Thanks, this and other typos fixed now.

    Regarding the intuitive picture: sure, intuitively it’s obvious, we simply have a base point •\bullet, a neighbour •∼∘\bullet \sim \circ and a neighbour of that •∼∘∼∘\bullet \sim \circ \sim \circ.

    (Beware remark 3.9 which alerts you to think of one of the two infinitesiaml points in a point of T Σ ∞ΣT^\infty_\Sigma \Sigma) as parameterizing where you are in the base (namely possibly infinitesimally close to a global point of the base). The other is the point in the infinitesimal disk around that point in the base.

    Michael Bachthold above in #8 is of course right that in general this neighbours-of-neighbours increases infinitesimal degree, but if one works at a given finite degree then there is a projection at the end that projects it back to that finite degree. One sees it from the explicit formula in example 3.16.

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 15th 2016
    • (edited Oct 15th 2016)

    Proposition 3.14.1 has X where it should be V.

    Remark 3.40 “autotmatically”

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeOct 16th 2016
    • (edited Oct 16th 2016)

    Does this operation define a vector bundle structure synthetically?

    By which I mean, can you prove synthetically that it is a vector bundle, not does it turn out to be a vector bundle in some model.

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 16th 2016
    • (edited Oct 16th 2016)

    Mike,

    Based on one of the Cockett-Crutwell Tangent category papers (arXiv:1606.08379), it’s likely you might get what they call a differential bundle, which is rather like a vector bundle, but only with the fibres looking like the underlying additive monoid. They do this entirely synthetically, and the SDG tangent bundle is an example, among others.

    [EDIT: added link above]

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeOct 16th 2016

    I’d be happy to get a bundle of abelian groups. In the purely synthetic world I don’t even see yet how to interpret T ΣT Σ→T ΣT_\Sigma T_\Sigma \to T_\Sigma as a binary operation, let alone an associative and commutative one.

    • CommentRowNumber17.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 17th 2016

    @Mike - looking at the paper, the definition of a differential bundle requires, up front, the map E→XE\to X to be a commutative monoid in the slice category, so I guess that’s already just an ε\varepsilon past what we have in the purely synthetic setting.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2016
    • (edited Oct 17th 2016)

    Mike,

    thanks for the prodding.

    So the general statement for the infinitesimal disk bundle monad T Σ ∞T^\infty_\Sigma induced from differential cohesion (ℜ⊣ℑ)(\Re \dashv \Im) is the following:

    1. For every XX, then T ∞XT^\infty X is a fiberwise monoid.

    2. For XX a VV-manifold, then T ∞XT^\infty X is a fiberwise group;

    3. For VV abelian, then T ∞XT^\infty X is fiberwise an abelian group.

    Specifically, for XX a VV-manifold, then the fiberwise monoid structure is just the restriction of the group structure on VV to its infinitesimal disk.

    I have written that out now as proposition 3.16 in the notes here.

    To see this, first notice that

    T Σ ∞T Σ ∞Σ≃(T ∞Σ)× Σ(T ∞Σ). T^\infty_\Sigma T^\infty_\Sigma \Sigma \simeq (T^\infty \Sigma) \times_\Sigma (T^\infty \Sigma) \,.

    You see this immediately from unwinding what it means that T Σ ∞T^\infty_\Sigma is the base change monad along Σ→ℑΣ\Sigma \to \Im \Sigma with T ∞Σ≔T Σ ∞ΣT^\infty \Sigma \coloneqq T^\infty_\Sigma \Sigma, or else see p. 19 of the notes here.

    This is what gives the monoid structure on T ∞ΣT^\infty \Sigma from the monad structure on T Σ ∞T^\infty_\Sigma.

    Next, use that the monad product is T Σ ∞evT^\infty_\Sigma \mathrm{ev} (prop. 3.15) and that over a VV-manifold then ev:T ∞Σ→Σev \colon T^\infty \Sigma \to \Sigma is locally the restriction of the group operation in VV along V×𝔻 e↪V×VV \times \mathbb{D}_e \hookrightarrow V \times V. (This is prop. 3.13, based on the abstract “Mayer-Vietoris argument” which you and me once discussed here, reflected on the nLab here).

    • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 17th 2016

    Ah, excellent, thanks Urs. Would there likely be some sort of weaker condition than being a VV-manifold, for VV an abelian group object, that might make the fibres of T ∞XT^\infty X commutative? Maybe some sort of distributive law??

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2016
    • (edited Oct 17th 2016)

    Sure, there might be weaker conditions, but I haven’t thought about them. As one does for microlinear spaces one could try to find a class of operations on spaces {X i}\{X_i\} under which the property is preserved that T ∞(−)T^\infty(-) is a fiberwise (abelian) group, if it is true for all the X iX_i.

    However, I don’t recall a crucial application for such microlinear spaces, beyond those locally modeled on a group VV (VV-manifolds). Is there an application you have in mind which you think needs to be dealt with?

    By the way, thanks for catching the typos in #13 ! I have fixed them now.

    • CommentRowNumber21.
    • CommentAuthorMichael_Bachtold
    • CommentTimeOct 17th 2016
    • (edited Oct 17th 2016)

    Thanks for the link to the writeup, Urs. Some questions (this refers to the version before you posted those last comments):

    Remark 3.9. p.20: wouldn’t a generalized point in T ∞XT^{\infty}X consist of just two generalized points in XX that are infinitesimally close? Where does the third point in that remark come from?

    Extending the intuitive picture of rem. 3.9 to the fibered case p:E→Xp:E\to X, is the following correct? A generalized point in T X ∞ET^{\infty}_X E consists of a point in ∘ 1:X\circ_1 : X and a point in ∘ 2:E\circ_2 : E sucht that the projection of ∘ 2\circ_2 to XX is infinitesimally close to ∘ 1\circ_1. The morphism π∘ev *p:T X ∞E→X\pi \circ \mathrm{ev}^{*}p : T^{\infty}_X E \to X simply discards the point ∘ 2\circ_2, while ev E:T X ∞E→E\mathrm{ev}_E: T^{\infty}_{X}E \to E discards ∘ 1\circ_1 (if this is correct, I don’t understand how ev E\mathrm{ev}_E can be seen as a morphism in H X\mathbf{H}_{X}, as written in prop. 3.11). Further, a generalized point in T X ∞T X ∞ET^\infty_{X} T^\infty_{X} E consists of two points in XX such that ∘ 1∼∘ 2\circ_1 \sim \circ_2 together with a third point ∘ 3:E\circ_3:E, such that its projection to XX is infinitesimally close to ∘ 2\circ_{2}. This seems to follow from the next commutative diagram, where all squares are pullbacks, and the right half is the same diagram as yours on page 19.

    T X ∞T X ∞E ⟶ T X ∞E ⟶ev E E ↓ ↓ ev *p ↓ p T ∞T ∞X ⟶ev T ∞X T ∞X ⟶ev X ↓ ↓ π ↓ η T ∞X ⟶ev X ⟶η ℑX ↓ π ↓ η X ⟶η ℑX \begin{matrix} T^{\infty}_{X}T^{\infty}_{X}E & \longrightarrow & T^{\infty}_{X}E & \overset{\mathrm{ev}_{E}}{\longrightarrow} & E\\ \downarrow & & \downarrow^{\scriptsize{\mathrm{ev}^{*}p}} & & \downarrow^{p}\\ T^{\infty}T^{\infty}X & \overset{\mathrm{ev}_{T^{\infty}X}}{\longrightarrow} & T^{\infty}X & \overset{\mathrm{ev}}{\longrightarrow} &X \\ \downarrow & & \downarrow^{\pi} & & \downarrow^{\eta}\\ T^{\infty}{X} & \overset{\mathrm{ev}}{\longrightarrow} & X & \overset{\eta}{\longrightarrow} & \Im X\\ \downarrow^{\pi} & & \downarrow^{\scriptsize{\eta}} & & \\ X &\overset{\eta}\longrightarrow & \Im X & & & \end{matrix}

    Isn’t the upper left horizontal arrow precisely the mondad product? If so, it should act by discarding the point in the middle, sending (∘ 1,∘ 2,∘ 3)(\circ_{1}, \circ_{2}, \circ_{3}) to (∘ 1,∘ 3) (\circ_{1},\circ_{3}).

    (Also, could you shed some light on why those arrows are called ev? I assume it is short for evaluation, but I don’t see how that is related.)

    I haven’t yet worked out at example 3.16 (now example 3.17), hope to come back to it.

    • CommentRowNumber22.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 17th 2016

    @Urs,

    no, just wondering if the class of objects XX for which T ∞XT^\infty X is likely to be a differential bundle a la Cockett-Crutwell is bigger than just the VV-manifolds for VV an abelian group object. I hadn’t thought about microlinear spaces. I guess I was more interested in providing C-C with more examples for their theory (which already encompasses ordinary tangent bundles, SDG tangent bundles, algebraic geometry and some stuff in computer science).

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2016
    • (edited Oct 17th 2016)

    @David, just to remark that the concept of “VV-manifolds” as in def 3.3 in the notes here is fairly general in its way. Effectively it says that all infinitesimal neighbourhoods in a space have group structure, and that these vary locally trivially. (And there is an implicit “∞\infty” sign everywhere…) At least intuitively this is just the minimum condition that one expects to need for tangent bundles to have the expected properties.

    • CommentRowNumber24.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 17th 2016

    @Urs, sure (and the ∞\infty-stuff was in my mind, too). One does wonder if the inverses in VV are needed for the basic results, but I guess without that you wouldn’t get triviality of T ∞VT^\infty V, or local triviality of T ∞XT^\infty X more generally, so it’s not exactly something you particularly want to look at!

    I just should sit down and see if I can show your VV-“manifold” synthetic tangent bundles are indeed examples of Cockett and Crutwell’s theory, for my own interest.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2016
    • (edited Oct 17th 2016)

    @Michal_Bachtold,

    yes, two points, each necessarily infinitesimal neighbours of a given global point (which you may think of as the image of either under X→ℑXX \to \Im X). There is a remark (now 3.9) which means to clarify this.

    As that remark also says, T ∞ΣT^\infty \Sigma is just what in algebraic geometry is called the formal neighbourhood of the diagonal, but thought of asymmetrically, in that one of the two projections out of the diagonal is now thought of as the bundle projection. The remaining projection is what is called “ev” now.

    This is for “evaluation”, because in the bundle perspective this is the map which takes a point in the fiber of the infinitesimal disk bundle and then “evaluates” it by sending it to the corresponding point in the base which it is identified with under the defining identification of the fiber with an infinitesimal disk in the base. If you think (as you may) of points in an infinitesimal disk as infinitesimal paths from the origin to that point, then this is the operation of “endpoint evaluation” in path spaces. Whence the name.

    Hence this is indeed not a morphism in H /X\mathbf{H}_{/X}. It may be regarded as a morphism in H /ℑX\mathbf{H}_{/\Im X}, if desired. Could you say again where in the file it says the former (in the new version)? I don’t see it presently, but that sounds like a typo. Thanks.

  26. I think it’s now prop 3.15 on p. 23. According to your last clarification should point 2 in that proposition read T ℑX ∞ev E:…T^\infty_{\Im X}\mathrm{ev}_E: \ldots?

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2016
    • (edited Oct 17th 2016)

    Oh, I see. No, that’s as stated, but with T ∞ΣT^\infty \Sigma sitting over Σ\Sigma via evev itself.

    I’ll highlight this clarification this in the file, thanks for alerting me. But not now, need to go offline now for a bit.

    • CommentRowNumber28.
    • CommentAuthorMichael_Bachtold
    • CommentTimeOct 17th 2016
    • (edited Oct 17th 2016)

    I see thanks. (Aside: you seem to be writing Σ\Sigma instead of XX in the document in several places. Not sure if you declared X=ΣX=\Sigma somewhere)

    Now, when you write

    T Σ ∞T Σ ∞Σ≃(T ∞Σ)× Σ(T ∞Σ). T^\infty_\Sigma T^\infty_\Sigma \Sigma \simeq (T^\infty \Sigma) \times_\Sigma (T^\infty \Sigma) \,.

    one could point out that the fibered product on the right is actually over two different projections from T Σ ∞ΣT^\infty_\Sigma \Sigma to Σ\Sigma. Does that really make the map (T ∞Σ)× Σ(T ∞Σ)→T ∞Σ(T^\infty \Sigma) \times_\Sigma (T^\infty \Sigma) \to T^\infty \Sigma a bundle of monoids?

    • CommentRowNumber29.
    • CommentAuthorMike Shulman
    • CommentTimeOct 17th 2016

    More later, but just one comment: microlinear spaces in SDG are closed under limits and exponentials, which classical manifolds are not. So for instance if MM is microlinear, so are End(M)End(M) and Aut(M)Aut(M), which are infinite-dimensional, and so is the zero-set of any function on a manifold, such as the cross {xy=0}⊆R×R\{x y = 0 \} \subseteq R\times R. I don’t immediately see why VV-manifolds should have such nice closure properties, even if we allow VV to vary.

    • CommentRowNumber30.
    • CommentAuthorMike Shulman
    • CommentTimeOct 17th 2016

    Also, before reading through this whole discussion, let me say that

    the fibered product on the right is actually over two different projections from T Σ ∞ΣT^\infty_\Sigma \Sigma to Σ\Sigma

    sounds like also what I would be worried about. Intuitively what I see is a groupoid (namely, the kernel pair of X→ℑXX\to \Im X), not a bundle of groups.

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2016

    Michael and Michael,

    you are right. I’ll fix it.

  32. (Just a thought on terminology: instead of calling those projections π\pi and ev, the names source and target could be also suggestive, maybe shortened to src, trg, or the like.)

  33. And would it be fair to call T ∞XT^\infty X the space of infinitesimal paths (or identifications) in XX?

    • CommentRowNumber34.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 17th 2017

    Were the worries of the two Michaels (#28 and #30) allayed by what Urs fixed (#31)?

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2017
    • (edited Oct 17th 2017)

    Remarkable, today is exactly one year ago that I made this silly mistake, which I like to think a more awake version of myself could not have commited. I had removed it immediately after it was pointed out. Let us not revive this annually! :-)

    • CommentRowNumber36.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 17th 2017

    But where does one end up without the silly mistake? People were having trouble understanding how the desired fibrewise monoid-group-abelian group structure is shown to exist synthetically.

    What are we left with after you fixed things?

    • CommentRowNumber37.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 17th 2017
    • (edited Oct 17th 2017)

    I guess I should be looking through p. 27 of v2 of Synthetic variational calculus (schreiber).

    If in 3.15, xx is an element in the base space XX (forgetting generalized elements for the moment), how is there a sum with an element, aa, of 𝔻 e×E\mathbb{D}_e \times E?

    I see that from aa and the projection, that such an evaluation can occur, but isn’t it strangely denoted?

    And then later in 3.16, why do we have (b,a,b)(b, a, b) in the iterated tangent space? Why the bb first, and how is that addition defined in (b,a+b)(b, a+b)?

    • CommentRowNumber38.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 17th 2017

    Typo: in the right hand term on line 4 of p. 28, part of it has slipped to subscript.

1 to 38 of 38

  1.