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- Discussion Type
- discussion topicsliced adjoint functors -- section
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jul 15th 2021

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.

- Discussion Type
- discussion topicfree loop space of classifying space
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by Ulrik
- Last Active Jul 15th 2021

am recording an actual proof that

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

- Discussion Type
- discussion topicflavour anomaly
- Category Latest Changes
- Started by Urs
- Comments 170
- Last comment by Urs
- Last Active Jul 15th 2021

- Discussion Type
- discussion topicEhrenfeucht-Fraïssé games
- Category Latest Changes
- Started by Ulrik
- Comments 4
- Last comment by Urs
- Last Active Jul 14th 2021

I’ve created the entry Ehrenfeucht-Fraïssé games and recorded the basic definitions and results (partly inspired by jesse’s efforts to add some model theory content to the nlab). I’m running out of steam now, but we should also mention the connections to quantifier elimination and Scott rank. (I’ve added a redirect from back-and-forth argument which was requested at quantifier elimination.)

- Discussion Type
- discussion topicDold-Kan correspondence
- Category Latest Changes
- Started by Urs
- Comments 28
- Last comment by Dmitri Pavlov
- Last Active Jul 14th 2021

added reference to dendroidal version of Dold-Kan correspondence

- Discussion Type
- discussion topicnonabelian cohomology
- Category Latest Changes
- Started by nLab edit announcer
- Comments 13
- Last comment by Urs
- Last Active Jul 14th 2021

- Discussion Type
- discussion topicfundamental theorem of dg-algebraic rational homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Jul 14th 2021

- Discussion Type
- discussion topiccosimplicial abelian group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 14th 2021

I am giving this its own little entry, for completeness and to go alongside

*simplicial abelian group*and*model structure on cosimplicial abelian groups*, and*monoidal Dold-Kan correspondence*

- Discussion Type
- discussion topicRational and p-adic Homotopy Theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 14th 2021

added missing cross-links with rational homotopy theory

- Discussion Type
- discussion topicfunction algebras on infinity-stacks
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 14th 2021

added full publication data for:

- Bertrand Toën,
*Champs affines*, Selecta Math. new series**12**(2006), no. 1, 39-135 (arXiv:math/0012219, doi:10.1007/s00029-006-0019-z)

- Bertrand Toën,

- Discussion Type
- discussion topicmodel structure on simplicial abelian groups
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Jul 14th 2021

I thought we already long had this as an entry – but, no, it was re-directing all along to

*model structure on simplicial algebras*. Am giving it its own entry now, but it remains telegraphic for the moment.

- Discussion Type
- discussion topicmodel structure on cosimplicial abelian groups
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 14th 2021

wrote model structure on cosimplicial abelian groups with a detailed proof that it is a simplicial model category.

This statement must be very well known, but it seems it is good to give an explicit proof.

- Discussion Type
- discussion topicSören Illman
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 13th 2021

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”

- Discussion Type
- discussion topiccellular approximation theorem
- Category Latest Changes
- Started by Tim_Porter
- Comments 11
- Last comment by Urs
- Last Active Jul 13th 2021

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?

- Discussion Type
- discussion topicequivariant Whitehead theorem
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Jul 13th 2021

- Discussion Type
- discussion topicG-CW complex
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Jul 13th 2021

started

*G-CW complex*.

- Discussion Type
- discussion topictransgression in group cohomology
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 13th 2021

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.

- Discussion Type
- discussion topicsimplicial set
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Jul 13th 2021

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.)

- Discussion Type
- discussion topictopological K-theory
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by nLab edit announcer
- Last Active Jul 13th 2021

started adding some basic technical details to topological K-theory

- Discussion Type
- discussion topiccylinder object
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Jul 13th 2021

I added to

*cylinder object*a pointer to a reference that goes through the trouble of spelling out the precise proof that for $X$ a CW-complex, then the standard cyclinder $X \times I$ is again a cell complex (and the inclusion $X \sqcup X \to X\times I$ 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 $X_n$, then the cells of $X_{n+1}$ glued in at top and bottom, then the further $(n+1)$-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. )

- Discussion Type
- discussion topicEilenberg-Zilber theorem
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Jul 13th 2021

created quick entry on Eilenberg-Zilber theorem

(to go with simplicial deRham complex)

but check. I am really in a hurry now and have to leave it in a somewhat stubby state.

- Discussion Type
- discussion topicalgebraic topology
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Jul 13th 2021

I have tried to give

*algebraic topology*a better Idea-section.

- Discussion Type
- discussion topicRocio Gonzalez-Diaz
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 13th 2021

brief

`category:people`

-entry for hyperlinking references at*Steenrod square*and at*Eilenberg-Zilber/Alexander-Whitney deformation retraction*

- Discussion Type
- discussion topicPedro Real
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 13th 2021

brief

`category:people`

-entry for hyperlinking references at*Steenrod square*and at*Eilenberg-Zilber/Alexander-Whitney deformation retraction*

- Discussion Type
- discussion topicdeformation retraction > history
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 13th 2021

This page did nothing but point to

*deformation retract*. Am have cleared this here now and instead installed a proper redirect

- Discussion Type
- discussion topicEilenberg-Zilber/Alexander-Whitney deformation retraction
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 13th 2021

- Discussion Type
- discussion topicfree abelian group
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Jul 12th 2021

wrote something at

*free abelian group*. Not great, need to come back to this.

- Discussion Type
- discussion topicbraid representation
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by nLab edit announcer
- Last Active Jul 12th 2021

- Discussion Type
- discussion topicpartial model category
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 3
- Last comment by Hurkyl
- Last Active Jul 12th 2021

Created:

## Idea

Partial model categories are one of the many intermediate notions between relative categories and model categories.

They axiomatize those properties of model categories that only involve weak equivalences.

## Definition

A

**partial model category**is a relative category such that its class of weak equivalences satisfies the 2-out-of-6 property (if $s r$ and $t s$ are weak equivalences, then so are $r$, $s$, $t$, $t s r$) and admits a 3-arrow calculus, i.e., there are subcategories $U$ and $V$ (which can be thought of as analogues of acyclic cofibrations and acyclic fibrations) such that $U$ is closed under cobase changes (which are required to exist), $V$ is closed under base changes, and any morphism can be functorially factored as the composition $v u$ for some $u\in U$ and $v\in V$.## Properties

If $(C,W)$ is a partial model category, then any Reedy fibrant replacement of the Rezk nerve $N(C,W)$ is a complete Segal space.

## Related concepts

## References

- Clark Barwick, Daniel M. Kan,
*Partial model categories and their simplicial nerves*, arXiv.

- Discussion Type
- discussion topicdivided power algebra
- Category Latest Changes
- Started by Théo de Oliveira S.
- Comments 19
- Last comment by Hurkyl
- Last Active Jul 12th 2021

- Discussion Type
- discussion topicminimal simplicial circle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 12th 2021

I am giving this its own little entry, in order to have a place for recording some basic facts and for hyperlinking at

*cyclic category*,*cyclic set*, etc., and also at*inertia orbifold*, etc.

- Discussion Type
- discussion topicsimplicial abelian group
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 12th 2021

The term

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

- Discussion Type
- discussion topicfunctor
- Category Latest Changes
- Started by Eric
- Comments 219
- Last comment by Urs
- Last Active Jul 12th 2021

Asked a question at functor.

- Discussion Type
- discussion topicdivision rig
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by nLab edit announcer
- Last Active Jul 12th 2021

- Discussion Type
- discussion topicdifferential linear logic
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 12th 2021

a stub, for the moment just so as to make links work at

*differential category*

- Discussion Type
- discussion topicJean-Simon Lemay
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 12th 2021

brief

`category:people`

-entry for hyperlinking references at*differential category*

- Discussion Type
- discussion topicdifferential category
- Category Latest Changes
- Started by Tim_Porter
- Comments 6
- Last comment by Tim_Porter
- Last Active Jul 12th 2021

- Discussion Type
- discussion topiccartesian differential category
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 7
- Last comment by Urs
- Last Active Jul 12th 2021

- Discussion Type
- discussion topicThomas Ehrhard
- Category Latest Changes
- Started by Tim_Porter
- Comments 4
- Last comment by Tim_Porter
- Last Active Jul 12th 2021

- Discussion Type
- discussion topicidempotent semiring
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 5
- Last comment by nLab edit announcer
- Last Active Jul 12th 2021

- Discussion Type
- discussion topicidempotent semifield
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 6
- Last comment by nLab edit announcer
- Last Active Jul 12th 2021

- Discussion Type
- discussion topicproduct of simplices
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Tim_Porter
- Last Active Jul 12th 2021

added pointer to:

- Greg Friedman, Section 5 of:
*An elementary illustrated introduction to simplicial sets*, Rocky Mountain J. Math. 42(2): 353-423 (2012) (arXiv:0809.4221, doi:10.1216/RMJ-2012-42-2-353)

- Greg Friedman, Section 5 of:

- Discussion Type
- discussion topicLaurent Regnier
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by Tim_Porter
- Last Active Jul 12th 2021

- Discussion Type
- discussion topicinertia orbifold
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Jul 12th 2021

- Discussion Type
- discussion topicRobin Cockett
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Jul 12th 2021

- Discussion Type
- discussion topicRichard Blute
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Jul 12th 2021

- Discussion Type
- discussion topicKurt Strebel
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 12th 2021

brief

`category:people`

-entry for hyperlinking references at*quadratic differential*

- Discussion Type
- discussion topicquadratic differential
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 12th 2021

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.

- Discussion Type
- discussion topicsimplicial mapping complex
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 11th 2021

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)

- Discussion Type
- discussion topicmodel structure on dg-algebras
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Urs
- Last Active Jul 11th 2021

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?

- Discussion Type
- discussion topicmonadic functor
- Category Latest Changes
- Started by Alec Rhea
- Comments 5
- Last comment by varkor
- Last Active Jul 11th 2021

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.

- Discussion Type
- discussion topicfunctor of descent type
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by varkor
- Last Active Jul 11th 2021

functor of descent type, not because I like the terminology but because it appears in the literature…

- Discussion Type
- discussion topicmonoidal model category
- Category Latest Changes
- Started by Urs
- Comments 19
- Last comment by Urs
- Last Active Jul 11th 2021

I have added to

*monoidal model category*statement and proof (here) of the basic statement:

Let $(\mathcal{C}, \otimes)$ 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 $(Ho(\mathcal{C}), \otimes^L, \gamma(I))$. If in in addition $(\mathcal{C}, \otimes)$ satisfies the monoid axiom, then 2) the localization functor $\gamma\colon \mathcal{C}\to Ho(\mathcal{C})$ carries the structure of a lax monoidal functor

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

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

- Discussion Type
- discussion topichigher inductive type
- Category Latest Changes
- Started by Urs
- Comments 77
- Last comment by Urs
- Last Active Jul 11th 2021

added pointer to

- Steve Awodey, Nicola Gambino, Kristina Sojakova,
*Homotopy-initial algebras in type theory*(arXiv:1504.05531)

to

*initial algebra over an endofunctor*,*higher inductive type*and*W-type*- Steve Awodey, Nicola Gambino, Kristina Sojakova,

- Discussion Type
- discussion topicGeoff Cruttwell
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Jul 11th 2021

- Discussion Type
- discussion topiclens (in computer science)
- Category Latest Changes
- Started by David_Corfield
- Comments 43
- Last comment by BryceClarke
- Last Active Jul 11th 2021

- Discussion Type
- discussion topicclosed bicategory
- Category Latest Changes
- Started by Mike Shulman
- Comments 7
- Last comment by BryceClarke
- Last Active Jul 11th 2021

- Discussion Type
- discussion topictransgression
- Category Latest Changes
- Started by Eric
- Comments 86
- Last comment by Urs
- Last Active Jul 10th 2021

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

- Discussion Type
- discussion topicinfinity-action
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by Ulrik
- Last Active Jul 10th 2021

started an entry

*infinity-action*

- Discussion Type
- discussion topicshuffle
- Category Latest Changes
- Started by Noam_Zeilberger
- Comments 2
- Last comment by Urs
- Last Active Jul 10th 2021