Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory itex k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes science set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 18th 2022

    Included the abstract of the article.

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 18th 2022

    Since someone was asking about this on Twitter, I tried to make some sense of Section II there. Is there more going on?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2022

    Interesting, I don’t remember appreciating that Section II before: it is re-casting the operations that go into the statement of the Hadamard lemma.

    Compare to Hadamard’s difference equation here:

    • The xx there (think of it as x0x - 0) is Lawvere’s Δx\Delta x

    • the f(x)f(0)f(x) - f(0) there is Lawvere’s Δy\Delta y,

    • and gg is gg.

    I’ll add cross-link.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2022

    added to the entry a brief comment on what happens in Section II: here

    diff, v5, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 18th 2022

    I was trying to make up a story for the enquirer about the adjoint cylinder aspect. So the composites pr i *diag *pr_i^{\ast} \circ \diag^{\ast} are both idempotent, but does it help to see a two variable function, g(x,y)g(x,y), as sandwiched between g(x,y)=g(x,x)g'(x,y) = g(x,x) and g(x,y)=g(y,y)g''(x,y)= g(y,y)? Some kind of extremes in class of two variable functions with same diagonal?

    Why am I reminded of that thing about intrinsic curvature? What is it? At any point, two perpendicular directions of max and min curvature.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 18th 2022

    Oh, is it infinitesimal cohesion, quality type terrain?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2022

    It seems to me that the relation to cohesion in this article is superficial.

    After all, that “cylinder” (here) is just the beginning of a cosimplicial ring, not an adjoint triple between categories. Even if one were promote it to an adjoint triple, say by extension/restriction of scalars between categories of modules, then the difference operation (“Δ\Delta in the notation of that section II) still is – while crucial for the argument – not accounted for by the diagrammatics, nor is its appearance in the formulas.

    What I see in the article is an example of taking an elementary fact, here the Hadamard lemma, and checking which bare-bones category theoretic structure is needed to formulate it, hence possibly to internalize it elsewhere. And here it’s not much: We need functions depending on either one or two variabables, and the ways to turn these into each other.

    To the extent that all this is meant to be about teaching kids (as the article seems to be suggesting), one could argue that it motivates the notion of Cartesian product and of their images under contravariant functors hiding in the formulation of a simle but profound classical lemma.

    Now Hadamard’s lemma is a deep statement about smooth differential geometry, I have called it the third of the three magical properties of the smooth category (here) – as such it underlies much of the construction and discussion of the canonical models of synthetic/cohesive differential geometry. But that deep property is not what the discussion in “Unity and Identity…” illuminates. It just appeals to it.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 18th 2022

    Ok, thanks. Perhaps no deep insight by Marx then:

    Near the end of his life, Karl Marx wrote about the foundations of differential calculus. The essence of his line of thought, later rigorously established by Hadamard, yields an effective and simple basis for learning and developing the subject if made explicit.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2022

    As I said, Hadamard’s lemma in itself is one of these simple lemmas that are crucially important (like the Yoneda lemma in category theory). If Marx really saw this that would be impressive. It would be fun to see this referenced. But maybe Lawvere got carried away here with attributing insight to Marx?

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2022

    If someone ever wishes to follow this up, Marx’s Mathematical Manuscripts.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)