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    • have created enriched bicategory in order to help Alex find the appropriate page for his notes.

    • 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 0→A 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 n≥0n\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

    • Created:

      Idea

      In algebraic geometry, the module of Kähler differentials of a commutative ring RR corresponds under the Serre–Swan duality to the cotangent bundle of the Zariski spectrum of RR.

      In contrast, the module of Kähler differentials of the commutative real algebra of smooth functions on a smooth manifold MM receives a canonical map from the module of smooth sections of the cotangent bundle of MM that is quite far from being an isomorphism.

      An example illustrating this point is M=RM=\mathbf{R}, since in the module of (traditionally defined) Kähler differentials of C ∞(M)C^\infty(M) we have d(exp(x))≠expdxd(exp(x))\ne exp dx, where exp:R→R\exp\colon\mathbf{R}\to\mathbf{R} is the exponential function. That is to say, the traditional algebraic notion of a Kähler differential is unable to deduce that exp′=exp\exp'=\exp using the Leibniz rule.

      However, this is not a defect in the conceptual idea itself, but merely a failure to use the correct formalism. The appropriate notion of a ring in the context of differential geometry is not merely a commutative real algebra, but a more refined structure, namely, a C^∞-ring.

      This notion comes with its own variant of commutative algebra. Some of the resulting concepts turn out to be exactly the same as in the traditional case. For example, ideals of C^∞-rings and modules over C^∞-rings happen to coincide with ideals and modules in the traditional sense. Others, like derivations, must be defined carefully, and definitions that used to be equivalent in the traditional algebraic context need not remain so in the context of C^∞-rings.

      Observe that a map of sets d:A→Md\colon A\to M (where MM is an AA-module) is a derivation if and only if for any real polynomial f(x 1,…,x n)f(x_1,\ldots,x_n) the chain rule holds:

      d(f(a 1,…,a n))=∑ i∂f∂x i(x 1,…,x n)dx i.d(f(a_1,\ldots,a_n))=\sum_i {\partial f\over\partial x_i}(x_1,\ldots,x_n) dx_i.

      Indeed, taking f(x 1,x 2)=x 1+x 2f(x_1,x_2)=x_1+x_2 and f(x 1,x 2)=x 1x 2f(x_1,x_2)=x_1 x_2 recovers the additivity and Leibniz property of derivations, respectively.

      Observe also that ff is an element of the free commutative real algebra on nn elements, i.e., R[x 1,…,x n]\mathbf{R}[x_1,\ldots,x_n].

      If we now substitute C^∞-rings for commutative real algebras, we arrive at the correct notion of a derivation for C^∞-rings:

      A __C^∞-derivation__ of a [[C^∞-ring]] $A$ is a map of sets $A\to M$ (where $M$ is a [[module]] over $A$) such that the following chain rule holds for every smooth function $f\in\mathrm{C}^\infty(\mathbf{R}^n)$:
$$d(f(a_1,\ldots,a_n))=\sum_i {\partial f\over\partial x_i}(x_1,\ldots,x_n) dx_i,$$
where both sides use the structure of a [[C^∞-ring]] to evaluate a smooth real function on a collection of elements in $A$.

      The module of Kähler C^∞-differentials can now be defined in the same manner as ordinary Kähler differentials, using C^∞-derivations instead of ordinary derivations.

      \begin{theorem} (Dubuc, Kock, 1984.) The module of Kähler C^∞-differentials of the C^∞-ring of smooth functions on a smooth manifold MM is canonically isomorphic to the module of sections of the cotangent bundle of MM. \end{theorem}

      Related concepts

      References

      v1, current

    • Categories enriched over groupoid form strict (2,1) categories. Edited for clarity.

      Mark Williams

      diff, v4, current

    • I strongly disagree with the statement in Grothendieck category that the Grothendieck category is small. The main examples like RMod{}_R Mod are not! What did the writer of that line have in mind ?

    • I added to the “abstract nonsense” section in free monoid a helpful general observation on how to construct free monoids. “Adjoint functor theorem” is overkill for free monoids over SetSet.

    • brief category:people-entry for hyperlinking references

      v1, current

    • starting stub on gaseous vector spaces

      Anonymouse

      v1, current

    • I have created lax morphism, with general definitions and a list of examples. It would be great to have more examples.

    • Created a stub page for this concept, which surprisingly didn’t exist yet.

      v1, current

    • starting article on set truncations

      Anonymouse

      v1, current

    • copying text from HoTT wiki

      Anonymous

      v1, current

    • I have added some things to frame. Mostly duplicating things said elsewhere (at locale and at (0,1)-topos), but I need these statements to be at frame itself.

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

      Anonymous

      v1, current

    • Clarify that the impredicative definition only quantifies over truth values.

      diff, v18, current

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

      Anonymous

      v1, current

    • At overt space there was a remark that since the definition quantifies over “spaces”, the overtness of a single space might depend on the general meaning chosen for “space”, but that no example was known to the author. I added an example involving synthetic topology, which may not be quite what the author of that remark was thinking of, but which I think is interesting.

    • starting something – not done yet but need to save

      v1, current

    • There has GOT to be a better photograph than that! Is there anyone here in Oxford? Can they go and get a picture for us?

    • Treated the case of comodules over corings, which differs from the definition for internal comonoids, though coring is an internal monoid. Namely, comodules over corings are defined as modules with coaction, rather than bimodules with coaction.

      diff, v16, current

    • I made some very minor changes to the introduction at descent. I hesitate to do more but at present the discussion does not seem that readable to me. Can someone look at it to see what they think? The intro seems to plunge in deep very quickly and so the ‘idea’ of descent as that of gluing local information together, does not come across to me. The article is lso quite long and perhaps needs splitting up a bit.

    • Added some content to display map from Taylor’s book. Not very deep, mostly as a reference to the respective section for me.

    • added a bare minimum sentence to the (previously empty) Idea-section, and more references

      diff, v4, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • The adeles are not the Cartesian product, but the restricted product.

      Reed

      diff, v15, current

    • Fixup description of birkhoff duality. Add small section on free distributive lattices.

      Gershom

      diff, v19, current

    • Created a page Morava K-theory . A lot to add. Will fill out later, with better reference list. Please edit!
    • In codomain fibration one calls the function

      C \ (-) : C --> Cat

      mapping c to the slice category (C \ c) a pseudofunctor. However I fail to see how this is not functorial.

      A morphism f : a --> b is sent to the functor (C \ f) : (C \ a) --> (C \ b) defined by (g : c --> a) |--> (fg : c --> b), and this assignment clearly satisfies composition. It also preserves identity. So what am I missing here?

    • Mathematician of the turn of 19/20th century.

      v1, current

    • have added more references to classical monographs

      diff, v16, current

    • I added the HoTT introduction rule for ’the’, then added a speculative remark on why say things like

      The Duck-billed Platypus is a primitive mammal that lives in Australia.

    • Spelled out the (equivalent) definition of locally small indexed category, and noted the equivalence of 2-categories explicitly.

      Eigil Rischel

      diff, v6, current

    • tried to bring the entry Lie group a bit into shape: added plenty of sections and cross links to other nLab material. But there is still much that deserves to be done.

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