Not signed in (Sign In)

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 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 homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory 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 nforum 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 set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft 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).

nLab > Latest Changes

Show all

13621 to 13680 of 14449

  1. <
  2. 1
  3. ...
  4. 224
  5. 225
  6. 226
  7. 227
  8. 228
  9. 229
  10. 230
  11. 231
  12. ...
  13. 241
  14. >

    • An Anonymous Coward changed a codecogs picture to an SVG at locale of real numbers, but it doesn’t look right to me, so I changed it back. (The SVG editor didn’t recognise it; otherwise, I’d have tried to fix it that way.)

    • There’s a note on how to do this at preset.

    • recorded at end in a new section Set-enriched coends as colimits the isomorphism

      ∫ d∈DW(d)⋅F(d)≃lim →((elW) op→D→FC). \int^{d \in D} W(d) \cdot F(d) \simeq \lim_{\to}( (el W)^{op} \to D \stackrel{F}{\to} C ) \,.

    • added characterizations of smooth kk-algebras to smooth scheme.

      Some expert please look at that and its relation to the rest of the entry.

    • I am trying to write up an elementary exposition for how the Hochschild chain complex for a commutative associate algebra is the normalized chains/Moore complex of the simplicial algebra that one gets by tensoring the algebra AA with the simplicial set Δ[1]/∂Δ[1]\Delta[1]/\partial \Delta[1]:

      C •(A,A)=N •((Δ[1]/∂Δ[1])⋅A). C_\bullet(A,A) = N_\bullet( (\Delta[1]/\partial \Delta[1]) \cdot A ) \,.

      I would like to get feedback on whether or not my exposition is in fact understandable in an elementary way.

      The section that contains this material is the section

      The simplicial circle algebra

      at the entry Hochschild cohomology. Just this one section. It’s not long.

      It describes first the simplicial set Δ[1]/∂Δ[1]\Delta[1]/\partial \Delta[1], then discusses how the coproduct in CAlg kCAlg_k is given by the tensor product over kk, and deduces from that what the simplicial algebra (Δ[1]/∂Δ[1])(\Delta[1]/\partial \Delta[1]) is like.

      After taking the normalized chains of that, the result is Pirashvili’s construction of a chain complex from a simplicial set and a commutative algebra. I just think it is important to amplify that this construction of Pirashvili’s is a categorical tensoring=copower operation. Because that connects the construction to general abstract constructions. That’s what the beginning of the above entry is about. But for the moment I would just like to make the elementary exposition of the tensoring operation itself pretty and understandable.

    • tried to bring the old neglected entry sSet-category roughly into some kind of stubby shape. Added Porter-Cordier and LurieA.3 as references. The former was my motivation for doing this. Eventually it would be good to have here a detailed discussion of sSetsSet-category models for (∞,1)(\infty,1)-category theory. See the discussion with Tim over in the other thread on the (∞,1)(\infty,1)-Yoneda lemma.

      (I don’t have time for this now. I am saying all this in the hope that somebody else has.)

    • I’ve cleaned up diffeological space a little. In particular:

      1. I’ve removed all references to Chen spaces. There is a relationship, but not what was implied on that page.
      2. I’ve tried to clean up the distinction between the definition in the literature (which uses all open subsets of Euclidean spaces) and the preferred nLab definition (which uses CartSp).
      3. Other minor cleaning.

    • I have been advising Herman Stel on his master thesis, which is due out in a few days. I thought it would be nice to have an nLab entry on the topic of the thesis, and so I started one: function algebras on infinity-stacks.

      For TT any abelian Lawvere theory, we establish a simplicial Quillen adjunction between model category structures on cosimplicial TT-algebras and on simplicial presheaves over duals of TT-algebras. We find mild general conditions under which this descends to the local model structure that models ∞\infty-stacks over duals of TT-algebras. In these cases the Quillen adjunction models small objects relative to a choice of a small full subcategory C⊂TAlg opC \subset T Alg^{op} of the localization

      L↪←LH=Sh (∞,1)(C) \mathbf{L} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} \mathbf{H} = Sh_{(\infty,1)}(C )

      of the (∞,1)(\infty,1)-topos of (∞,1)(\infty,1)-sheaves over duals of TT-algebras at those morphisms that induce isomorphisms in cohomology with coefficients the canonical TT-line object. In as far as objects of H\mathbf{H} have the interpretation of ∞-Lie groupoids the objects of L\mathbf{L} have the interpretatin of ∞-Lie algebroids.

      For the special case where TT is the theory of ordinary commutative algebras this reproduces the situation of (Toën) and many statements are straightforward generalizations from that situation. For the case that TT is the theory of smooth algebras (C ∞C^\infty-rings) we obtain a refinement of this to the context of synthetic differential geometry.

      As an application, we show how Anders Kock’s simplicial model for synthetic combinatorial differential forms finds a natural interpretation as the differentiable ∞\infty-stack of infinitesimal paths of a manifold. This construction is an ∞\infty-categorical and synthetic differential resolution of the de Rham space functor introduced by Grothendieck for the cohomological description of flat connections. We observe that also the construction of the ∞\infty-stack of modules lifts to the synthetic differential setup and thus obtain a notion of synthetic ∞\infty-vector bundles with flat connection.

      The entry is of course as yet incomplete, as you will see.

    • I created split coequalizer and absolute coequalizer, the latter including a characterization of all absolute coequalizers via an “nn-ary splitting.” While I was doing this, I noticed that monadic adjunction included a statement of the monadicity theorem without a link to the corresponding page, so I added one. (The discussion at the bottom of monadic adjunction should probably be merged into the page somehow.) Then I noticed that while we had a page preserved limit, we didn’t have reflected limit or created limit, so I created them. They could use some examples, however.

      I would also like to include an example of how to actually use the monadicity theorem to prove that a functor is monadic. Something simpler than the classic example in CWM about compact Hausdorff spaces; maybe monadicity of categories over quivers? Probably not something that you would need the monadicity theorem for in practice, so that it can be simple and easy to understand.

    • Under the Definitions at topos, I gave definitions of Grothendieck toposes and W-toposes, since these are two very important kinds of toposes that some authors (at least) often call simply ‘toposes’. (Also it gave me a place to redirect W-topos and its synonym topos with NNO.)

    • On Tim’s suggestion, I have been in contact with Ronnie Brown on some of the history behind “convenient categories”, and received a wealth of information from him. I have made an initial attempt to summarize what I have learned in the Historical Remarks section of convenient category of topological spaces, but it might be somewhat garbled still. Hopefully Ronnie and/or Tim will have a look. I will be adding more references by and by.

      I also added in the follow-up discussion with Callot under Counterexamples.

    • Francois Métayer already has a lab-entry. (NB the acute accent is missing on the new one! That was the cause of the error.)

    • I added some more details to the section on ultrafilter monad at ultrafilter. Incidentally, it seems to me that the bit on Barr’s observation (“topological spaces = relational β\beta-modules”) is too terse. There is a lot of generalized topology via abstract nonsense that deserves more explanation.

      I also added some elementary material to cartesian closed category, mainly to indicate to the novice how exponentials deserve to be thought of as function spaces, how internal composition works, and so forth. I left the job somewhat unfinished.

    • Krzysztof Worytkiewicz kindly informs me that an old question to Francois Metayer has now been answered: the folk model structure on strict ω\omega-categories does restrict to the Brown-Golasinski model structure on strict ω\omega-groupoids. (The latter is indeed the transferred model structure along the forgetful functor to the former).

      This is now written up in

      Ara, Métayer, The Brown-Golasinki model structure on oo-groupoids revisited (pdf)

      As my connection allows, I will insert this into the nLab entry now…

    • New stub hom-connection. I should figure it out once. While tensor product is involved in many constructions in algebra, some are dual with Hom instead, for example there are contramodules in addition to comodules over a coring. In similar vain hom-connections were devised, but there are some really intriguing examples (including superconnections, right connections of Manin etc.) and there are relations to examples of noncommutative integration of various kind.

    • I have added some discussion to the page on orientals (in the sense of Ross Street), regarding the link to the convex geometry of cyclic polytopes (as discussed by Kapranov and Voevodsky).

      My selfish motive for doing so is that I am curious if my recent work with Steffen Oppermann which includes a new description of the triangulations of (even-dimensional) cyclic polytopes, has any relevance to the study of orientals, or higher category theory more broadly. (In particular, if there are explicit questions about the internal structure of orientals which are of interest, I would like to hear about them.)

      A particularly speculative version of my question, would be whether there is a natural connection between orientals and the representation theory which we are studying in that paper (which necessitated a detour into convex geometry). We biject triangulations of an even-dimensional cyclic polytope to (a nice class of) tilting objects for a certain algebra. The simplest version of this (which was already known) is that triangulations of an nn-gon correspond to tilting objects for the path algebra of the quiver consisting of a directed path with n−2n-2 vertices. (These tilting objects then give derived equivalences between the derived category of this path algebra, and the derived category of the endomorphism ring of this tilting object.)

      Questions, speculations, or suggestions would be very welcome.

      Hugh

    • to the functional analysis crew of the nnLab: where should operator spectrum point to? Do we have any suitable entry?

    • New stubs Oka principle, Oka manifold (with redirect Oka map) and Franc Forstnerič. Jardine has shown that one can use the Toen-Vezzosi like engineering with his intermediate model structure on the category of simplicial presheaves on a simplicial version of the Stein site. The (∞,1)(\infty,1)-stacks/fibrants will be Oka maps; those cofibrants which are represented by complex manifolds are in fact Stein manifolds.

    • I expanded some entries related to the Café-discussion:

      • at over-(infinity,1)-topos I expanded the Idea-section, added a few remarks on proofs and polished a bit,

        and added the equivalence ∞Grpd/X≃PSh ∞(X)\infty Grpd/X \simeq PSh_{\infty}(X) to the Examples-section

      • at base change geometric morphism I restructured the entry a little and then included the proof of the existence of the base change geometric morphism

    • added to adjunct the description in terms of units and counits.

    • created (infinity,1)-algebraic theory.

      I tried to adapt Rosicky’s and Lurie’s terminology such as to match that at algebraic theory, but Mike, Toby, Todd and whoever else feels expert should please check if I did it right.

    • added the equivalence

      Top/X≃Sh (∞,1)(X) Top/X \simeq Sh_{(\infty,1)}(X)

      here

    • Kevin Walker was so kind to add a bit of material to blob homology. Notably he added a link to a set of notes now available that has more details.

      I added formatting and some hyperlinks.

    • I added to loop space a reference to Jim’s classic article, which was only linked to from H-space and put pointers indicating that his delooping result in TopTop is a special case of a general statement in any ∞\infty-topos.

      By the way: it seems we have slight collision of terminology convention here: at “loop space” it says that H-spaces are homotopy associative, but at “H-space” only a homotopy-unital binary composition is required, no associativity. I think this is the standard use. I’d think we need to modify the wording at loop space a little.

    • I reworked A-infinity algebra so as to apply to algebras over any A ∞A_\infty-operad in any ambient category. So I created subsections “In chain complexes”, “In topological spaces”.

      I think if we speak generally of “algebra over an operad” then we should also speak generally of “A ∞A_\infty-algebra” even if the enriching category is not chain complexes. Otherwise it will become a mess. But I did link to A-infinity space.

    • added the definition of “coloured operad” to operad in the section “Rough definition”

      (by the way, should we not rather call these “pedestrian definition” or so instead of “rough”? The latter seems to suggest that there is something not quite working yet with these definitions, while in fact they are perfectly fine, just not as high-brow as other definitions.)

13621 to 13680 of 14449

  1. <
  2. 1
  3. ...
  4. 224
  5. 225
  6. 226
  7. 227
  8. 228
  9. 229
  10. 230
  11. 231
  12. ...
  13. 241
  14. >
Top of Page