added pointer to:

- Emily Riehl,
*Could $\infty$-category theory be taught to undergraduates?*, Notices of the AMS (May 2023) [published pdf, arxiv:2302.07855]

added pointer to:

- Emily Riehl,
*$\infty$-Category theory for undergraduates*, talk at*CQTS*(Dec. 2022) [video: rec]

Changed “full sub-quasicategory” to “wide sub-quasicategory”in the definition of “relative quasicategory”.

]]>added publication data and hyperlink to:

- Emily Riehl, Dominic Verity,
*Elements of ∞-Category Theory*, Cambridge studies in advanced mathematics**194**, Cambridge University Press (2022) $[$doi:10.1017/9781108936880, ISBN:978-1-108-83798-9, pdf$]$

repaired outdated link to Moritz Groth A short course on $\infty$-categories

]]>Fixed the link (it needs the `https://`

prefix in order to point outside of the domain `ncatlab.org`

).

I have also updated links accordingly at *infinity-cosmos*, at *homotopy 2-category of quasi-categories* and at *Emily Riehl*.

Updated website for Elements of $\infty$-Category Theory

]]>added this pointer:

- Markus Land,
*Introduction to Infinity-Categories*, Birkhäuser 2021 (doi:10.1007/978-3-030-61524-6)

added publication data for this item in the list of references:

- Julie Bergner,
*A survey of $(\infty,1)$-categories*, In: John Baez, Peter May (eds.),*Towards Higher Categories*The IMA Volumes in Mathematics and its Applications, vol 152. Springer, New York, NY (arXiv:math/0610239, doi:10.1007/978-1-4419-1524-5_2)

What’s the latest that one could cite regarding survey of the web of Quillen equivalences between the different models for $(\infty,1)$-categories?

Let’s see…

Here is one, sort of:

- Julia Bergner,
*Equivalence of models for equivariant $(\infty,1)$-categories*, Glasgow Mathematical Journal, Volume 59, Issue 1 (2016) (arXiv:1408.0038, doi:10.1017/S0017089516000136)

added publication data to

- Denis-Charles Cisinski,
*Higher category theory and homotopical algebra*, Cambridge University Press 2019 (doi:10.1017/9781108588737, pdf)

Added the updated reference:

Thanks!!

]]>There is an old version on Doug Ravenel’s webpages: https://web.math.rochester.edu/people/faculty/doug/otherpapers/Riehl-Verity-ICWM.pdf but I do not know its status.

On Emily’s home pages is a reference and link to http://www.math.jhu.edu/~eriehl/elements.pdf which is probably the best source for the new version. There is already a link to this on the n-Lab page.

]]>Added the updated reference:

- Emily Riehl and Dominic Verity,
*Elements of $\infty$-Category Theory*, (2019) (pdf)

The pdf link in

- Emily Riehl and Dominic Verity,
*$\infty$-Categories for the Working Mathematician*, (2018) (pdf)

is broken, and (I seem to remember) the project was renamed, too. So I am removing this line hereby. But if you know which pointer should go here instead, please add it.

]]>Added

- Emily Riehl,
*Homotopical categories: from model categories to (∞,1)-categories*(arXiv:1904.00886)

added pointer to

- Emily Riehl,
*The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories*, talk at Vladimir Voevodsky Memorial Conference 2018 (pdf)

Added reference to Riehl-Verity’s book.

]]>have added pointer to

- Emily Riehl, Dominic Verity,
*Infinity category theory from scratch*, 2016 (pdf)

Has anything more been made of the other approach to stratified spaces where one moves up through strata and back down again? You may remember that discussion at the Cafe here. It gave rise to Transversal homotopy theory by Jon Woolf, who also wrote a paper mentioned by Ayala and Rozenblyum, The fundamental category of a stratified space.

The idea was to give fundamental categories with duals.

]]>Added pointer to the new preprint by Ayala and Rozenblyum. Though it doesn’t seem to have the previously announced statement about $(\infty,n)$-categories with duals yet.

]]>Thanks for catching that, I have fixed the sentence now and expanded it such as to read as follows:

More precisely, this is the notion of *category* up to coherent homotopy:
an $(\infty,1)$-category is equivalently

an internal category in ∞-groupoids/basic homotopy theory (as such usually modeled as a complete Segal space).

a category homotopy enriched over ∞Grpd (as such usually modeled as a Segal category).

Is an (∞,1)-category really an internal category in ∞-groupoids, as that would mean the object of objects would be an ∞-groupoid,

Yes, it’s the completeness condition of complete Segal spaces that takes care of this issue. Details are at *internal category in an (∞,1)-category*.

Yes, but is that completely correct? The ‘enriched’ version to me was clearer. Is an (∞,1)-category really an internal category in ∞-groupoids, as that would mean the object of objects would be an ∞-groupoid, or am I mistaken?

]]>It was probably supposed to be “an internal category in …” or “a category internal to …”.

]]>In the entry on (infinity,1)-category there is the phrase:an (∞,1)-category is an internal to in ∞-groupoids/basic homotopy theory.

I tried to see how to clear up the grammar, but it was not clear to me what the wording was intended to be. There was a previous version:

To some extent an (∞,1)-category can be thought of as a category enriched in (∞,0)-categories, namely in ∞-groupoids.

That is vague, so needed changing, but there seem to be ’typos’ in the current version.

]]>