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- Discussion Type
- discussion topicEilenberg-Zilber map
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 10th 2021

added pointer to:

- Samuel Eilenberg, Joseph A. Zilber, American Journal of Mathematics Vol. 75, No. 1 (Jan., 1953), pp. 200-204 (jstor:2372629, doi:10.2307/2372629)

- Discussion Type
- discussion topicepimorphism in an (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 10th 2021

See also the discussion here.

- Discussion Type
- discussion topicdiagonal of a bisimplicial set
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 10th 2021

added

some more formatting;

cross-links with

*total complex*,*totalization*,*total simplicial set*etc.pointers to:

Antonio Cegarra, Josué Remedios,

*The relationship between the diagonal and the bar constructions on a bisimplicial set*, Topology and its applications, volume 153 (1) (2005) (pdf, doi:10.1016/j.topol.2004.12.003)Danny Stevenson, pages 2 & 11 of:

*Décalage and Kan’s simplicial loop group functor*, Theory and Applications of Categories, Vol. 26, 2012, No. 28, pp 768-787 (arXiv:1112.0474, tac:26-28)

- Discussion Type
- discussion topiceffective epimorphism in an (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by Urs
- Last Active Jul 10th 2021

I have split off effective epimorphism in an (infinity,1)-category from effective epimorphism and polished and expanded slightly.

- Discussion Type
- discussion topicBin Zhang
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 10th 2021

brief

`category:people`

-entry for hyperlinking references at*orbifold K-theory*and elsewhere

- Discussion Type
- discussion topic2-group
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Jul 10th 2021

I looked at the entry

*2-group*today and found it strongly wanting. Now I have spent a few minutes with it, trying to bring it into better shape. While I think I did imporve it a little, there is clearly still lots of further room for improvement.Mainly what I did was add more on the intrinsic meaning and definition, more on the homotopical meaning, and more on the details of how crossed modules present the $(2,1)$-category of 2-groups – amplifying the role of weak equivalences. And brief remarks on how all this generalizes to the case of 2-groups “with structure” hence internal to other $\infty$-toposes than the terminal one.

- Discussion Type
- discussion topicNora Ganter
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by Urs
- Last Active Jul 10th 2021

- Discussion Type
- discussion topicPlatonic 2-group
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Jul 10th 2021

Finally I am starting an entry

*Platonic 2-group*.For the moment, all it has is the statement of Epa-Ganter 16, prop. 4.1, rephrased as the diagram

$\array{ \mathcal{G}_{uni}[i] &\longrightarrow& \mathbf{B}G_{ADE} &\longrightarrow& \mathbf{B}^3 \mathbb{Z}/{\vert G_{ADE}\vert} \\ \downarrow && \downarrow && \downarrow \\ String(SU(2)) &\longrightarrow& \mathbf{B} SU(2) &\underset{\mathbf{c}_2}{\longrightarrow}& \mathbf{B}^3 U(1) }$

- Discussion Type
- discussion topichomotopy product
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 6
- Last comment by Urs
- Last Active Jul 9th 2021

- Discussion Type
- discussion topicsymmetric bicategory
- Category Latest Changes
- Started by varkor
- Comments 8
- Last comment by Urs
- Last Active Jul 9th 2021

- Discussion Type
- discussion topicfinite type
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 8th 2021

- Discussion Type
- discussion topicobject of finite type
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by Urs
- Last Active Jul 8th 2021

- Discussion Type
- discussion topicinductive family
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by Urs
- Last Active Jul 8th 2021

- Discussion Type
- discussion topicsequential limit
- Category Latest Changes
- Started by Hurkyl
- Comments 1
- Last comment by Hurkyl
- Last Active Jul 8th 2021

Included the condition on sequential (co)limits that the indexing ordinal should be nonzero, which I presume to be the correct convention. (e.g. based on the description they are a special case of filtered colimits)

- Discussion Type
- discussion topic2-category of adjunctions
- Category Latest Changes
- Started by David_Corfield
- Comments 14
- Last comment by varkor
- Last Active Jul 7th 2021

- Discussion Type
- discussion topicWeil algebra
- Category Latest Changes
- Started by Urs
- Comments 31
- Last comment by jim_stasheff
- Last Active Jul 7th 2021

edited Weil algebra a bit. More to come.

- Discussion Type
- discussion topicelementary (infinity,1)-topos
- Category Latest Changes
- Started by David_Corfield
- Comments 87
- Last comment by David_Corfield
- Last Active Jul 7th 2021

I added the reference

- Nima Rasekh,
*A Theory of Elementary Higher Toposes*, (arXiv:1805.03805)

- Nima Rasekh,

- Discussion Type
- discussion topicWilliam Dwyer
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 6
- Last comment by Tim_Porter
- Last Active Jul 7th 2021

- Discussion Type
- discussion topicPL de Rham complex of smooth manifold is equivalent to de Rham complex
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 7th 2021

- Discussion Type
- discussion topicWilliam G. Dwyer > history
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 3
- Last comment by Urs
- Last Active Jul 7th 2021

- Discussion Type
- discussion topicrestriction of scalars
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 7th 2021

- Discussion Type
- discussion topiccategory of fibrant objects
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by nLab edit announcer
- Last Active Jul 6th 2021

in reaction to an email discussion I had, I have finally added to the section

*Derived hom-spaces*at*category of fibrant objects*the definition and theorem that had been alluded to there all along.

- Discussion Type
- discussion topicJeremy Gibbons
- Category Latest Changes
- Started by BryceClarke
- Comments 1
- Last comment by BryceClarke
- Last Active Jul 6th 2021

- Discussion Type
- discussion topicDynkin diagram
- Category Latest Changes
- Started by Grant_Bradley
- Comments 1
- Last comment by Grant_Bradley
- Last Active Jul 6th 2021

- Discussion Type
- discussion topicsimplicial skeleton
- Category Latest Changes
- Started by mattecapu
- Comments 6
- Last comment by Urs
- Last Active Jul 6th 2021

- Discussion Type
- discussion topicderivator
- Category Latest Changes
- Started by Urs
- Comments 98
- Last comment by Urs
- Last Active Jul 6th 2021

Added to derivator the explanation that Denis-Charles Cisinski had posted to the blog.

Zoran, I have made the material you had here the section "References", as this was mainly pointers to the literature. Please move material that you think you should go into other sections.

- Discussion Type
- discussion topicPlancherel measure
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Dmitri Pavlov
- Last Active Jul 6th 2021

- Discussion Type
- discussion topicKakeya Conjecture
- Category Latest Changes
- Started by nLab edit announcer
- Comments 9
- Last comment by felipe
- Last Active Jul 6th 2021

- Discussion Type
- discussion topicexternalization
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Jul 6th 2021

- Discussion Type
- discussion topicSerre spectral sequence
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Jul 6th 2021

I have added also the statement of the two relative versions of the Serre spectral sequence (here). No details yet.

- Discussion Type
- discussion topicMartin Bojowald
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 6th 2021

brief

`category:people`

-entry for hyperlinking this reference on structure formation during cosmic inflation via inhomogeneous quantum cosmology:- Martin Bojowald, Ding Ding,
*Canonical description of cosmological backreaction*, JCAP**03**(2021) 083 (arXiv:2011.03018)

- Martin Bojowald, Ding Ding,

- Discussion Type
- discussion topiccategorical model of dependent types
- Category Latest Changes
- Started by Mike Shulman
- Comments 40
- Last comment by David_Corfield
- Last Active Jul 6th 2021

Created categorical model of dependent types, describing the various different ways to strictify category theory to match type theory and their interrelatedness. I wasn’t sure what to name this page — or even whether it should be part of some other page — but I like having all these closely related structures described in the same place.

- Discussion Type
- discussion topicinhomogeneous cosmology
- Category Latest Changes
- Started by Urs
- Comments 32
- Last comment by Urs
- Last Active Jul 6th 2021

started something. For the moment really just a glorified pointer to Buchert et al. 15 and putting Scharf 13 into perspective

- Discussion Type
- discussion topicMathieu Anel
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Jul 6th 2021

Added reference

- Mathieu Anel,
*The elementary infinity-topos of truncated coherent spaces*(arXiv:2107.02082)

- Mathieu Anel,

- Discussion Type
- discussion topichomotopy type with finite homotopy groups
- Category Latest Changes
- Started by David_Corfield
- Comments 13
- Last comment by David_Corfield
- Last Active Jul 6th 2021

- Discussion Type
- discussion topicMealy morphism
- Category Latest Changes
- Started by nLab edit announcer
- Comments 9
- Last comment by BryceClarke
- Last Active Jul 5th 2021

- Discussion Type
- discussion topiccofibration category
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jul 5th 2021

Added these references:

Karol Szumiło, Homotopy theory of cofibration categories. Homology Homotopy Appl. 18 (2016), no. 2, 345–357. doi

Karol Szumiło, Homotopy theory of cocomplete quasicategories. Algebr. Geom. Topol. 17 (2017), no. 2, 765–791. doi

Krzysztof Kapulkin, Karol Szumiło, Quasicategories of frames of cofibration categories. Appl. Categ. Structures 25 (2017), no. 3, 323–347. doi

Karol Szumiło, Frames in cofibration categories. J. Homotopy Relat. Struct. 12 (2017), no. 3, 577–616. doi

Markus Land, Thomas Nikolaus, Karol Szumiło, Localization of cofibration categories and groupoid C∗-algebras. Algebr. Geom. Topol. 17 (2017), no. 5, 3007–3020. doi

- Discussion Type
- discussion topictrivial group
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 5th 2021

I tweaked the text at

*trivial group*a little.

- Discussion Type
- discussion topicsimplicial classifying space
- Category Latest Changes
- Started by Urs
- Comments 30
- Last comment by Urs
- Last Active Jul 5th 2021

- Discussion Type
- discussion topicfinite rotation group
- Category Latest Changes
- Started by Urs
- Comments 42
- Last comment by Urs
- Last Active Jul 5th 2021

I am splitting off an entry

*classification of finite rotation groups*from*ADE classification*in order to collect statements and references specific to the classification of finite subgroups of $SO(3)$ and $SU(2)$.Is there a canonical reference for the proof of the classification statement? I find lots of lecture notes that give the proof, but all of them without citing sources or original publications of proofs.

- Discussion Type
- discussion topicicosahedral group
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Urs
- Last Active Jul 5th 2021

added to

$\array{ \mathcal{I} &\hookrightarrow& String_{SU(2)} \\ \downarrow && \downarrow \\ 2 I &\hookrightarrow & Spin(3) = SU(2) \\ \downarrow && \downarrow \\ I \simeq A_5 &\hookrightarrow& SO(3) \\ \downarrow && \downarrow \\ I_h \simeq A_5\times \mathbb{Z}/2 &\hookrightarrow & O(3) }$*icosahedral group*discussion of the distinction of definitions as one moves up the Whitehead tower of $O(3)$[edit: added analogous discussion to

*octahedral group*and*icosahedral group*]

- Discussion Type
- discussion topicoctahedral group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 5th 2021

added pointer to Section 4.2 of:

- Satoshi Tomoda, Peter Zvengrowski,
*Remarks on the cohomology of finite fundamental groups of 3-manifolds*, Geom. Topol. Monogr. 14 (2008) 519-556 (arXiv:0904.1876)

for computation of the group cohomology

- Satoshi Tomoda, Peter Zvengrowski,

- Discussion Type
- discussion topictetrahedral group
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 5th 2021

- Discussion Type
- discussion topicPeter Zvengrowski
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 5th 2021

brief

`category:people`

-entry for hyperlinking references at*finite rotation group*

- Discussion Type
- discussion topicSatoshi Tomoda
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 5th 2021

brief

`category:people`

-entry for hyperlinking references at*finite rotation group*

- Discussion Type
- discussion topicquaternion group
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Jul 5th 2021

added the statement (here) that of all finite subgroups of $SU(2)$, $Q_8$ is a proper subgroup of the three exceptional ones.

Checking normality of this subgroup, I noticed that there is an issue with another item of the entry here, where it used to claim that a finite group is Hamiltonian precisely of it “contains a copy of $Q_8$”. But this can’t be, can it. I changed it to saying that every Hamiltonian group contains $Q_8$ as a subgroup, which I suppose is what was meant.

[edit: I see now that the statement that I changed back to was made already by Thomas Holder in rev 3, while the statement I removed was made by Thomas in rev 4. Thomas, if you see this, please let me know. ]

- Discussion Type
- discussion topichomotopy coherent nerve
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Hurkyl
- Last Active Jul 5th 2021

added to homotopy coherent nerve two diagrams in the section Examples and illustrations that are supposed to illustrate the hom-SSets of the simplicial category on

- Discussion Type
- discussion topicBryce Clarke
- Category Latest Changes
- Started by BryceClarke
- Comments 3
- Last comment by BryceClarke
- Last Active Jul 5th 2021

- Discussion Type
- discussion topicdelta lens
- Category Latest Changes
- Started by BryceClarke
- Comments 2
- Last comment by BryceClarke
- Last Active Jul 4th 2021

Created a page for delta lens containing a general idea, definition, some examples, and key references.

- Discussion Type
- discussion topiclocal epimorphism
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by nLab edit announcer
- Last Active Jul 4th 2021

I have

touched the formatting of this ancient entry,

expanded and streamlined the Idea-section a little,

added illustrating diagrams for the definition of the sieves appearing in the definition (here) of local epis from a given site

added the version of the definition for covergages instead of Grothendieck topologies (here, is this in Johnstone?)

added statement of the proposition that the Cech nerve projection out of a local epi is a local weak equivalence of simplicial presheaves (here)

- Discussion Type
- discussion topicoptic (in computer science)
- Category Latest Changes
- Started by David_Corfield
- Comments 6
- Last comment by RodMcGuire
- Last Active Jul 4th 2021

- Discussion Type
- discussion topicBorel construction
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Jul 4th 2021

started an entry on the

*Borel construction*, indicating its relation to the nerve of the action groupoid.

- Discussion Type
- discussion topicsimplicial group action
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 4th 2021

for completeness, to go alongside

*topological G-space*etc., but also for*Borel model structure*

- Discussion Type
- discussion topicgroup object
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Jul 4th 2021

Added to group object the Yoneda-embedding-style definition and added supergroup to the list of examples.

- Discussion Type
- discussion topicalgebra for a C-C bimodule
- Category Latest Changes
- Started by SridharRamesh
- Comments 21
- Last comment by Mike Shulman
- Last Active Jul 4th 2021

I've created a new article entitled algebra for a C-C bimodule, a straightforward concept encapsulating both algebras and coalgebras for endofunctors, as well as further generalities besides. There's surely a better name than using "C-C bimodule" (replacing it with "endoprofunctor", perhaps? Although I actually find that less preferable...), for someone to propose or let me know already exists, as the case may be.

(I've also made some small edits to the articles on algebras and coalgebras for endofunctors; in particular, the former had forgotten to define the morphisms of such algebras)

- Discussion Type
- discussion topicmodule object
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Jul 4th 2021

expanded the Idea-section and added a reference:

- Francis Borceux, George Janelidze, Gregory Maxwell Kelly, p. 8 of:
*Internal object actions*, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 (dml:249553)

- Francis Borceux, George Janelidze, Gregory Maxwell Kelly, p. 8 of:

- Discussion Type
- discussion topicmodel structure on reduced simplicial sets
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Jul 4th 2021

have created model structure on reduced simplicial sets

(I thought I had a vague memory that this or something similarly titled already existed, but apparently it didn’t).

Also added a little bit more detail on the Quillen equivalence with simplicial groups here and there, notably in the last section at groupoid object in an (infinity,1)-category

- Discussion Type
- discussion topicright/left Kan fibration
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Jul 4th 2021

- Discussion Type
- discussion topicsimplicial group
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by Urs
- Last Active Jul 4th 2021

at simplicial group I added/expanded the section delooping and simplicial principal bundles

I discuss this in revisionistic terms meant to exhibit the simple general underlying structure, and then try to spell out how it corresponds to vaarious explicit constructions in the literature, trying to point out page and verse in May’s “Simplicial objects in algebraic topology” and discuss how that yields what I am discussing.

By the way, did anyone ever find the time to make a sanity check of my query-box claim at decalage that forming decalage in sSet is nothing but forming the standard based path space object?

- Discussion Type
- discussion topicCalculus of fractions and homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 4th 2021

I have added reference to Gabriel-Zisman’s

*Calculus of fractions and homotopy theory*to the following entries:and maybe others. Usually, I’d conversely create a hyperlinked index at

*Calculus of fractions and homotopy theory*, but no time right now.