Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• am giving this its own little entry, since there are several entries that need to refer to this statement, so that it’s good to have a single place where to collect all the cross-links and references (of which I have only one, for the moment)

• Prop. 2.13 “stable” should be “strict”

Anonymous

• Created a stub for cofunctor? with some references.

• edited decalage a bit

there was the statement that $Dec Y \to Y$ is a "fibration". I made that Kan fibration. Is that right?

• there was a pointer to the generalization of “faithful functor” to 2-categories. I have added below that pointer to the corresponding version for $(\infty,1)$-categories.

• I need a word for the homotopy quotient $(\mathcal{L}X)/S^1$ of free loop spaces $\mathcal{L}X$ by their canonical circle action. It seems that the only term in use with respect to this is “twisted loop space”, which however usually refers just to the constant loops $(\mathcal{L}_{const}X)//S^1$. Since under nice conditions the derived functions on the $\mathcal{L}Spec(A)/S^1$ is the cyclic homology complex of $A$, I suggest that a good name is “cyclic loop space”. I made a quick note at cyclic loop space, just to fix and disambiguate terminology.

• brief category:people-entry for hyperlinking references at cyclic loop space

• brief category:people-entry for hyperlinking references at cyclic loop space

• starting something, not done yet

• Created stub to clear grey link.

Roman T

• made some minor cosmetic edits, such as replacing

  \bar W G


(which comes out with too short an overline) with

  \overline{W} G

• for the moment just to make links work

• An attempt to create this page was made by Paulo Perrone, but the creation was not successful. Am creating the page without any content beyond ’TODO’ now as a test.

• a stub, to satisfy links

• In the examples section of extensive category, it is stated that the category of affine schemes is infinitary extensive.

For all I know, I was the one who stuck in that example. But is that statement actually true? I’m having trouble seeing it.

If $S$ is a commutative ring over $R$ (by which I mean under $R$ (-:), does the functor $S \otimes_R -: CAlg_R \to CAlg_R$ preserve arbitrary cartesian products? Because it seems that’s what we basically need for the statement to be true.

• Since it was mentioned by Urs on g+, I thought I’d start mysterious duality. Maybe not a great name when someone discovers how it works (as someone claims to have done here).

• am giving this its own entry, for ease of hyperlinking. Made up the terminology, maybe there is a better choice.

Anonymous

• Rewrote misleading comment seeming to imply wide pullbacks need to be preserved to get preservation of equalizers.

Kevin Carlson

• Added the contents of the canonical isomorphism induced by some non-canonical isomorphism as coming from Lack’s proof.

• a bare sub-section with a list of references – to be !included into relevant entries – mainly at confinement and at mass gap problem (where this list already used to live)

• starting something. Not done yet but need to save

• Have added to cyclic set a pointer to notes from 1996 by Ieke Moerdijk where the theory classified by the topos of cyclic sets is identified (abstract circles).

This is an unpublished note, but on request I have now uploaded it to the nLab

• Ieke Moerdijk, Cyclic sets as a classifying topos, 1996 (pdf)

I have also added a corresponding brief section to classifying topos.

By the way, there is an old query box with an exchange between Mike and Zoran at cyclic set. It seems to me that this has been resolved and the query box could be removed (to make the entry read more smoothly). Maybe Mike and/or Zoran could briefly look into this.

• Fixed pdf link to “Towards an understanding of Girard’s transcendental syntax”

ALH

• stub for free loop space (in $Top$)

• added the definition to cyclic homology

next the task is to write out the details for how under the identification of the Hochschild complex with functions on the derived loop space, the cyclic complex is the $S^1$-equivariant functions on the derived loop space.

• I have begun cleaning up the entry cycle category, tightening up definitions and proofs. This should render some of the past discussion obsolete, by re-expressing the intended homotopical intuitions (in terms of degree one maps on the circle) more precisely, in terms of “spiraling” adjoints on the poset $\mathbb{Z}$.

Here is some of the past discussion I’m now exporting to the nForum:

The cycle category may be defined as the subcategory of Cat whose objects are the categories $[n]_\Lambda$ which are freely generated by the graph $0\to 1\to 2\to\ldots\to n\to 0$, and whose morphisms $\Lambda([m],[n])\subset\mathrm{Cat}([m],[n])$ are precisely the functors of degree $1$ (seen either at the level of nerves or via the embedding $\mathrm{Ob}[n]_\Lambda\to \mathbf{R}/\mathbf{Z}\cong S^1$ given by $k\mapsto k/(n+1)\,\mathrm{mod}\,\mathbf{Z}$ on the level of objects, the rest being obvious).

The simplex category $\Delta$ can be identified with a subcategory of $\Lambda$, having the same objects but with fewer morphisms. This identification does not respect the inclusions into $Cat$, however, since $[n]$ and $[n]_\Lambda$ are different categories.

• Added a reference to Guitart’s early paper.

• added a reference to Dynamical Systems and Sheaves by Schultz, Spivak, and Vasilakopoulou

Joe M

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

• Created semi-simplicial set, mainly as a repository for some terminological remarks. I would welcome anyone more knowledgeable about the history to correct or improve it!

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

• Created.

• brief category:people-entry for hyperlinking references

• I have changed the title of this article, as well as references to the object within it. Use of the term “Hawaiian Earring” is objected to by Hawaiian mathematicians. Please see these two threads, one by native Hawaiian and math PhD Dr. Marissa Loving, and the other by an expert on the Hawaiian Earring, Dr. Jeremy Brazas.