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    • added to group cohomology

      • in the section structured group cohomology some remarks about how to correctly define Lie group cohomology and topological group cohomology etc. and how not to

      • in the section Lie group cohiomology a derivation of how from the right oo-categorical definition one finds after some unwinding the correct definition as given in the article by Brylinski cited there.

      it's late here and I am now in a bit of a hurry to call it quits, so the proof I give there may need a bit polishing. I'll take care of that later...

    • Hi Dmitri, I see you just made “pointed homotopy class” redirect here. But should it not redirect to “homotopy class”, since it’s about maps, not about spaces? I’ll add a line there to highlight the pointed case.

      diff, v5, current

    • Added the statement on existence of right adjoints that only requires the target to be locally small.

      diff, v9, current

    • Added some more words to this entry, mentioning the homotopy category and pointed homotopy classes.

      diff, v4, current

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

      v1, current

    • Removed the following discussion to the nForum:

      Zoran Škoda: But there is much older and more general theorem of Hurewitz: if one has a map p:EBp:E\to B and a numerable covering of BB such that the restrictions p 1(U)Up^{-1}(U)\to U for every UU in the covering is a Hurewicz fibration then pp is also a Hurewicz fibration. But the proof is pretty complicated. For example George Whitehead’s Elements of homotopy theory is omitting it (page 33) and Postnikov is proving it (using the equivalent “soft” homotopy lifting property).

      Todd Trimble: Yes, I am aware of it. You can find a proof in Spanier if you’re interested. I’ll have to check whether the Milnor trick (once I remember all of it) generalizes to Hurewicz’s theorem.

      Stephan: I wonder if this trick moreover generalizes (in a homotopy theoretic sense) to categories other that Top\Top; for example to the classical model structure on CatCat?

      diff, v7, current

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

      v1, current

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

      v1, current

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

      v1, current

    • Based on a private discussion with Mike Shulman, I have added some explanatory material to inductive type. This however should be checked. I have created an opening for someone to add a precise type-theoretic definition, or I may get to this myself if there are no takers soon.

    • added pointer to the original reference (as kindly supplied by Dmitri over at Lie derivative):

      • Élie Cartan, Leçons sur les invariants intégraux (based on lectures given in 1920-21 in Paris, Hermann, Paris 1922, reprinted in 1958).

      Also cleaned up some text in this entry here.

      diff, v6, current

    • I am unhappy with Lie derivative. In the previous version it defined the Lie derivative as a secondary notion, using the differential and the Cartan homotopy formula (for which I finally created an entry). I have added a bit mentioning vector fields etc. and a formula using derivatives for forms but this is still not the right thing. Namely, in my understanding the Lie derivative is a fundamental notion and should not be defined using other differential operators, but by the “fisherman’s derivative” formula. Second it makes sense not only for differential forms but for any geometric quantities associated to the (co)frame bundle, and in particular to any kind of tensors, not necessarily contravariant or antisymmetrized. For this one has a prerequisite which will require some work in nnLab. Namely to a vector field, one associated the flow, not necessarily defined for all times, but for small times. Then for any tt one has a diffeomorphism, which is used in the fisherman’s formula. But fisherman’s formula requires the pullback and the pullback is usually defined for forms while for general tensor fields one may need combination of pullbacks and pushforwards. However, for diffeomorphisms, one can define pullback in both cases, and pullback for time tt flow corresponds to the pushforward for time t-t. To define such general pullback it is convenient to work with associated bundles for frame or coframe bundle and define it in the formalism of associated bundles. In the coframe case, this is in Sternberg’s Lectures on differential geometry (what returns me back into great memories of the summer 1987/1988 when I studied that book). So there is much work to do, to add details on those. If somebody has comments or shortcuts to this let me know.

      However, there is a scientific question here as well: what about when frame bundle is replaced by higher jet bundles, and one takes some higher differential operator for functions and wants to do a similar program – are there nontrivial extensions of Lie derivative business to higher derivatives which does not reduce to the composition of usual Lie derivatives ?

    • Added a reference:

      A coordinate-free treatment first appeared in

      • Harley Flanders, Development of an extended exterior differential calculus. Transactions of the American Mathematical Society 75:2 (1953), 311–311. doi.

      diff, v11, current

    • fix wrong definition of free group action

      Alexey Muranov

      diff, v32, current

    • I’ve removed this query box from metric space and incorporated its information into the text:

      Mike: Perhaps it would be more accurate to say that the symmetry axiom gives us enriched \dagger-categories?

      Toby: Yeah, that could work. I was thinking of arguing that it makes sense to enrich groupoids in any monoidal poset, cartesian or otherwise, since we can write down the operations and all equations are trivial in a poset. But maybe it makes more sense to call those enriched \dagger-categories.

    • Discovered this old stub entry and boosted it up a little.

      diff, v4, current

    • at Serre fibration I have spelled out the proof that that with F xXfibYF_x \hookrightarrow X \overset{fib}{\to} Y then π (F)π (X)π (Y)\pi_\bullet(F) \to \pi_\bullet(X)\to \pi_{\bullet(Y)} is exact in the middle. here.

      (This is intentionally the low-technology proof using nothing but the definition. )

    • I made the former entry "fibered category" instead a redirect to Grothendieck fibration. It didn't contain any addition information and was just mixing up links. I also made category fibered in groupoids redirect to Grothendieck fibration

      I also edited the "Idea"-section at Grothendieck fibration slightly.

      That big query box there ought to be eventually removed, and the important information established in the discussion filled into a proper subsection in its own right.

    • touched the formatting in congruence, fixed a typo on the cartesian square, added a basic example

    • Having encountered the third reference here, I was looking to find out what it was about, and got as far as this stub.

      Seems to tie in interestingly with fracture squares and localization, so probably some modal HoTT about.

      v1, current

    • Removing an old discussion:

      Is there a reason that you moved these references up here? We need them especially for the stuff about morphisms below. —Toby


      Eric: What would a colimit over an MSet-valued functor F:AMSetF:A\to MSet look like?

      Toby: That depends on what the morphisms are.

      Eric: I wonder if there is enough freedom in the definition of morphisms of multisets so that the colimit turns out particularly nice. I’m hoping that it might turn out to be simply the sum of multisets. According to limits and colimits by example the colimit of a Set-valued functor is a quotient of the disjoint union.

      Toby: I think that you might hope for the coproduct (but not a general colimit) of multisets to be a sum rather than a disjoint union. Actually, you could argue that the sum is the proper notion of disjoint union for abstract multisets.

      diff, v38, current

    • Todd,

      you added to Yoneda lemma the sentence

      In brief, the principle is that the identity morphism id x:xxid_x: x \to x is the universal generalized element of xx. This simple principle is surprisingly pervasive throughout category theory.

      Maybe it would be good to expand on that. One might think that the universal property of a genralized element is that every other one factors through it uniquely. That this is true for the generalized element id xid_x is a tautological statement that does not need or imply the Yoneda lemma, it seems.

    • It was pointed out to me today that in the very special case of internal (0,1)-category objects in Set, what we are calling a “pre-category” reduces to a preordered set, while adding the “univalence/Rezk-completeness” condition to make it a “category” promotes it to a partially ordered set. I feel like surely I knew that once, but if so, I had forgotten. It provides some extra weight behind this term “precategory”, especially since some category theorists like to say merely “ordered set” to mean “partially ordered set”.

    • Added section on irreflexive comparisons, which generalises linear orders in constructive mathematics


      diff, v15, current

    • I added a short abstract description of the Cartesian fibration over the interval, and commented that the section describes a construction with additional strictness properties in the quasi-category model.

      diff, v43, current

    • Describing the arrangements which have been made for funding of the nLab in collaboration with the Topos Institute. The page, linked to from the home page, is intended to be fairly general; specific requests for donations can be made elsewhere.

      v1, current

    • A comparatively long and technical section “From hom-functors to units and counits” (on adjoint functors) was sitting inside the Idea-section of adjunction. It seemed plainly misplaced there, and distracting attention from what should be the content of this entry, as opposed to the entry adjoint functor. So I have moved it now to where it seems to belong: inside the Examples-section.

      diff, v55, current

    • starting something, in order to have a place where to record the splitting over vector bundles SESs over paracompact spaces.

      (Still need to add a more canonical reference…)

      v1, current

    • This is a short article with a definition of the category of filters, now called the category of filters.

      I apologise for vandalising by mistake the article on filters, and thanks to Richard Williamson for fixing this. I tried to “edit a current page .. in context on a relevant page” misunderstanding the intructions from HowTo:

      How to start a new page

      You do this in two steps, the first of which may have already been done:

      Create a preliminary link (represented by a question mark) by editing > a current page and putting the name of the new page in double square brackets


      v1, current

    • Have added to HowTo a description for how to label equations

      In the course of this I restructured the section “How to make links to subsections of a page” by giving it a few descriptively-titled subsections.

    • added brief mentioning of the vertical tangent bundle and statement of the splitting formula

      TP(π *TB) P(T πP) T P \;\simeq\; \big( \pi^\ast T B \big) \oplus_P \big( T_\pi P \big)

      diff, v11, current