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    • this is a bare little section, to be !include-ed as a Properties-subsection at slice category and adjoint (∞,1)-functor (where two copies of this same section used to be all along) and also at slice (∞,1)-category and adjoint functor (where, for completeness, the same should be recorded, too, but wasn’t until now)

      It would also be good to expand a little here, for instance by adding a pointer to a 1-category textbook account (this is probably in Borceux, but I haven’t checked yet), or, of course, by adding some indication of the proof.

      v1, current

    • am recording an actual proof that

      (W¯𝒢)𝒢 ad𝒢 \mathcal{L} \big( \overline{W}\mathcal{G} \big) \;\; \simeq \;\; \mathcal{G}_{ad} \sslash \mathcal{G}

      I expected that a proof for this folklore theorem would be citeable from the literature, but maybe not quite. This MO reply points to Lemma 9.1 in arXiv:0811.0771, which has the idea (in topological spaces), but doesn’t explicitly verify all ingredients. I have tried to make it fully explicit (in simplicial sets).

      v1, current

    • am starting some minimum here. Have been trying to read up on this topic. This will likely become huge towards beginning of next year

      v1, current

    • I added a reference to a paper of mine

      Amnon Yekutieli

      diff, v48, current

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

      v1, current

    • added earlier publication items:

      • Sören Illman, Equivariant singular homology and cohomology for actions of compact lie groups (doi:10.1007/BFb0070055) In: H. T. Ku, L. N. Mann, J. L. Sicks, J. C. Su (eds.), Proceedings of the Second Conference on Compact Transformation Groups Lecture Notes in Mathematics, vol 298. Springer 1972 (doi:10.1007/BFb0070029)

      • Sören Illman, Equivariant algebraic topology, Annales de l’Institut Fourier, Tome 23 (1973) no. 2, pp. 87-91 (doi:10.5802/aif.458)

      and also more cross-links under “Related entries”

      diff, v5, current

    • Someone anonymous has raised the question of subdivision at cellular approximation theorem. I do not have a source here in which I can check this. Can anyone else check up?

    • fixed the pointers to alleged proofs of the equivariant Whitehead theorem.

      What is a canonical citation for this statement, for the unstable case, citable as an actual proof? Is it due to Bredon? Is it in his book?

      diff, v3, current

    • In this entry I mean to write out a full proof for the transgression formula for (discrete) group cocycles, using just basic homotopy theory and the Eilenberg-Zilber theorem.

      Currently there is an Idea-section and the raw ingredients of the proof. Still need to write connecting text. But have to interrupt for the moment.

      v1, current

    • added to simplicial set in the Definition section a slightly more explicit version of the definition.

      (I see now this kind of thing is repeated further below in the entry. But it should be right there as a formal definition, I think.)

    • I added to cylinder object a pointer to a reference that goes through the trouble of spelling out the precise proof that for XX a CW-complex, then the standard cyclinder X×IX \times I is again a cell complex (and the inclusion XXX×IX \sqcup X \to X\times I a relative cell complex).

      What would be a text that features a graphics which illustrates the simple idea of the proof, visualizing the induction step where we have the cylinder over X nX_n, then the cells of X n+1X_{n+1} glued in at top and bottom, then the further (n+1)(n+1)-cells glued into all the resulting hollow cylinders? (I’d like to grab such graphics to put it in the entry, too lazy to do it myself. )

    • Since we had bits and pieces of this scattered around over several entries, I am giving this its own little entry, so as to have the statement and all relevant references collected in one place, for ease of updating/synchronizing and hyperlinking.

      v1, current

    • giving this its own little entry, for ease of hyperlinking

      v1, current

    • The term simplicial abelian group used to redirect to simplicial group. I am giving it its own little entry now, for better hyperlinking.

      v1, current

    • division rigs, the rig version of division rings

      Anonymous

      v1, current

    • Just a stub for the moment to try to introduce the notion of differential category due to Blute, Cockett and Seely.

      v1, current

    • Started writing down the definition.

      v1, current

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

      v1, current

    • Added some remark on the order of a semiring. Actually, does anybody know if any semiring embedds into a semifield?

      diff, v5, current

    • Created this entry with mention of lattice structure and a fundamental theorem of algebra for semifields.

      v1, current

    • added redirect plus some ‘selected publications’.

      diff, v2, current

    • added redirects for abbreviations of his name, plus some publications. Also added a (for the moment Grey link) to differential category, which hopefully I will be able to add a sutb for later.

      diff, v4, current

    • Found this abandoned stub entry from 2010. Have removed the line

      Zoran: This is just a reminder for me to work on this entry in few days (to do list)

      and instead added some minimum structure, including formatting, a line in the Idea-section (but not doing it justice) and some more references.

      diff, v2, current

    • some minimum, for the moment just for the convenience that the link works

      (in creating this entry I noticed that we have an ancient stub entry mapping complex that deserves some attention)

      v1, current

    • Fixed the comments in the reference list at model structure on dg-algebras: Gelfand-Manin just discuss the commutative case. The noncommutative case seems to be due to the Jardine reference. Or does anyone know an earlier one?

    • The link for ’equivalent’ at the top redirected to natural isomorphism which (as I understand it) is the correct 1-categorical version of an equivalence of functors, but this initially lead me to believe that a functor was monadic iff it was naturally isomorphic to a forgetful functor from the Eilenberg-Moore category of a monad on its codomain, which would mean that the domain of the functor was literally the Eilenberg-Moore category of some adjunction since natural isomorphism is only defined for parallel functors.

      diff, v19, current

    • I have added to monoidal model category statement and proof (here) of the basic statement:


      Let (𝒞,)(\mathcal{C}, \otimes) be a monoidal model category. Then 1) the left derived functor of the tensor product exsists and makes the homotopy category into a monoidal category (Ho(𝒞), L,γ(I))(Ho(\mathcal{C}), \otimes^L, \gamma(I)). If in in addition (𝒞,)(\mathcal{C}, \otimes) satisfies the monoid axiom, then 2) the localization functor γ:𝒞Ho(𝒞)\gamma\colon \mathcal{C}\to Ho(\mathcal{C}) carries the structure of a lax monoidal functor

      γ:(𝒞,,I)(Ho(𝒞), L,γ(I)). \gamma \;\colon\; (\mathcal{C}, \otimes, I) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(I)) \,.

      The first part is immediate and is what all authors mention. But this is useful in practice typically only with the second part.

    • I started an idea section at transgression, but it could probably use some going over by an expert. I hope I didn’t mess things up too badly. I was reading Urs’ note on “integration without integration” on the train ride home and fooled myself into thinking I understood something.

      By the way, this reminded me of a discussion we had a while back

      Integrals: Loops space vs target space