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    • created stub for Wick's lemma, for the moment just so as to record a pointer to a reference

    • stub, for the moment just so as to satisfy links

      v1, current

    • displayed coinductive types in dependent type theory

      v1, current

    • added to differential operator the characterization via bundle maps out of a jet bundle, together with the note that this means that differential operators are equivalently morphisms in the co-Kleisli category of the Jet bundle comonad.

    • Page created, but author did not leave any comments.

      v1, current

    • removing question in query box

      +– {: .query} Are there (necessarily nonmetrizable) complete uniform spaces that are not Baire spaces? =–

      Anonymous

      diff, v23, current

    • Page created, but author did not leave any comments.

      Anonymous

      v1, current

    • Page created, but author did not leave any comments.

      Anonymous

      v1, current

    • I have given the definition of (infinity,1)-pretopos an entry, from appendix A of Jacob Lurie’s “Spectral Algebraic Geometry”.

      Since this is defined via a variant of the Giraud-Rezk-Lurie axioms with some of the infinitary structure made finitary, and since there is a large supply of non-Grothendieck examples (the sub-\infty-categories of coherent objects in any Grothendieck \infty-topos are \infty-pretoposes), it would be interesting if \infty-pretoposes were examples of elementary (infinity,1)-toposes, for some definition of the latter.

      Are they?

    • Created:

      Background

      See the article Kähler C^∞-differentials of smooth functions are differential 1-forms for the necessary background for this article, including the notions of C^∞-ring, C^∞-derivation, and Kähler C^∞-differential.

      Idea

      In algebraic geometry, (algebraic) differential forms on the Zariski spectrum of a [commutative ring RR (or a commutative kk-algebra RR) can be defined as the free commutative differential graded algebra on RR.

      This definition does not quite work for smooth manifolds: as already explained in the article Kähler C^∞-differentials of smooth functions are differential 1-forms, the notion of a Kähler differential must be refined in order to extract smooth differential 1-forms from the C^∞-ring of smooth functions on a smooth manifold MM.

      Thus, in order to get the algebra of smooth differential forms, the notion of a commutative differential graded algebra must likewise be adjusted.

      \begin{definition} A commutative differential graded C^∞-ring is a real commutative differential graded algebra AA whose degree 0 component A 0A_0 is equipped with a structure of a C^∞-ring in such a way that the degree 0 differential A 0A 1A_0\to A_1 is a C^∞-derivation. \end{definition}

      With this definition, we can recover smooth differential forms in a manner similar to algebraic geometry, deducing the following consequence of the Dubuc–Kock theorem for Kähler C^∞-differentials.

      \begin{theorem} The free commutative differential graded C^∞-ring on the C^∞-ring of smooth functions on a smooth manifold MM is canonically isomorphic to the differential graded algebra of smooth differential forms on MM. \end{theorem}

      Application: the Poincaré lemma

      The Poincaré lemma becomes a trivial consequence of the above theorem.

      \begin{proposition} For every n0n\ge0, the canonical map

      R[0]Ω(R n)\mathbf{R}[0]\to \Omega(\mathbf{R}^n)

      is a quasi-isomorphism of differential graded algebras. \end{proposition}

      \begin{proof} (Copied from the MathOverflow answer.) The de Rham complex of a finite-dimensional smooth manifold MM is the free C^∞-dg-ring on the C^∞-ring C (M)C^\infty(M). If MM is the underlying smooth manifold of a finite-dimensional real vector space VV, then C (M)C^\infty(M) is the free C^∞-ring on the vector space V *V^* (the real dual of VV). Thus, the de Rham complex of a finite-dimensional real vector space VV is the free C^∞-dg-ring on the vector space V *V^*. This free C^∞-dg-ring is the free C^∞-dg-ring on the free cochain complex on the vector space V *V^*. The latter cochain complex is simply V *V *V^*\to V^* with the identity differential. It is cochain homotopy equivalent to the zero cochain complex, and the free functor from cochain complexes to C^∞-dg-rings preserves cochain homotopy equivalences. Thus, the de Rham complex of the smooth manifold VV is cochain homotopy equivalent to the free C^∞-dg-ring on the zero cochain complex, i.e., R\mathbf{R} in degree 0. \end{proof}

      References

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • a stub, for the moment just to make some links work

      v1, current

    • I’ve made a few changes at flat module since I wanted to know what one was and the nLab page simply confused me further. It seemed to be saying that a module over a ring/algebra AA is flat if tensoring with AA is a flat functor. That seemed absurd so I changed it. The observation “everything happens for a reason” was a little curt so once I’d worked out what it meant I expanded it a bit and put in the analogy to bases. It wasn’t clear from the way that it was phrased whether this condition was due to Wraith and Blass or the fact that it can be put in a more general context. Lastly, the last sentence was originally in the same paragraph as the penultimate sentence where it didn’t seem to fit (or was at best ambiguous) and it also claimed that the module could be non-unital which seemed a little odd.

      If an expert could kindly check that I’ve done no lasting damage to the page, I’d be grateful.

    • The book is no longer in progress, but published 8 years ago. I added the detail and a link to the AMS page.

      I think it would be good to include a paragraph on the claims in the book about (,2)(\infty,2)-categories the authors explicitly say they don’t prove and can’t find a proof in the literature. Just flagging this for now. Ultimately, when the papers finishing the proofs of these claims land on the arXiv, these can be cited.

      diff, v4, current

    • Did some editing on this page. There’s a query there about whether the Grothendieck ring of a braided monoidal category is commutative. Seems so from here, so I’ll remove it.

      diff, v4, current

    • Page created, but author did not leave any comments.

      Anonymous

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • I added some material about monads and adjunctions in the 2-category Rel and decided to distinguish this 2-category from the 1-category of relations, hoping this will make it a bit easier to state lots of results about both without getting mixed up.

      diff, v23, current

    • added to canonical form references (talk notes) on canonicity or not in the presence of univalence

    • starting something.

      So far just a recounting of the statement of Prop. 25 in Serre 1977

      and some references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • removed link to old philosophy paper

      steveawodey

      diff, v9, current

    • I am taking the liberty of creating a category: reference-entry in order to have a way to hyperlink references to our new research center here in NYUAD, which is slowly but surely entering into tangible existence.

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • For all sections of

      for which there is a corresponding nLab entry (most of them) I have added this pointer to the entry.

      diff, v8, current

    • I worked on synthetic differential geometry:

      I rearranged slightly and then expanded the "Idea" section, trying to give a more comprehensive discussion and more links to related entries. Also added more (and briefly commented) references. Much more about references can probably be said, I have only a vague idea of the "prehistory" of the subject, before it became enshrined in the textbooks by Kock, Lavendhomme and Moerdijk-Reyes.

      Also, does anyone have an electronic copy of that famous 1967 lecture by Lawvere on "categorical dynamics"? It would be nice to have an entry on that, as it seems to be a most visionary and influential text. If I understand right it gave birth to topos theory, to synthetic differential geometry and all that just as a spin-off of a more ambitious program to formalize physics. If I am not mistaken, we are currently at a point where finally also that last bit is finding a full implmenetation as a research program.

    • I added more info on pseudo double categories and double bicategories to double category. I also simplified the picture of a square, which had been bristling with scary unnecessary detail. There's a slight blemish in the left vertical arrow, which I can't see how to fix.
    • seeing Eric create diffeology I became annoyed by the poor state that the entry diffeological space was in. So I spent some minutes expanding and editing it. Still far from perfect, but a step in the right direction, I think.

      (One day I should add details on how the various sites in use are equivalent to using CartSp)

    • a bare minimum, for the moment just to make the link work

      v1, current

    • Just to be clear, if at wrapped cycle ϕ *[Σ]\phi_\ast [\Sigma] is a multiple of a cycle cc, would we or wouldn’t we say it wrapped it?

    • brief category:people-entry for hyperlinking references

      v1, current

    • have a quick suggestion for a definition at embedding,

      This was the first idea that came to mind when reading Toby’s initial remark there, haven’t really thought much about it.

    • started page to make links work

      Anonymouse

      v1, current

    • added to shape theory a section on how strong shape equivalence of paracompact spaces is detected by oo-stacks on these spaces

      By the way: I have a question on the secion titled "Abstract shape theory". I can't understand the first sentence there. It looks like this might have been broken in some editing process. Can anyone fix this paragraph and maybe expand on it?

    • added pointer to

      • J. Montesinos, A representation of closed orientable 3-manifolds as 3-fold branched coverings of S 3S^3, Bull. Amer. Math. Soc. 80 (1974), 845-846 (Euclid:1183535815)

      here and also at 3-manifold and 3-sphere

      diff, v11, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • the entry braid group said what a braid is, but forgot to say what the braid group is; I added in a sentence, right at the beginning (and fixed some other minor things).

    • a stub, need the link to work, but nothing to be seen here just yet

      v1, current

    • Created a stub entry for norm map, for the moment just so as to make cross-links work.

    • More references including last week’s big paper.

      diff, v5, current