When pointing somebody to it, I noticed that the entry *n-category* is in a rather sad state and in particular it used to start out in a rather unhelpful fashion. I have now tried to briefly fix at least the latter problem by expanding and editing the first two sentences a bit. Notably I made sure that a pointer to *(∞,n)-category* appears early on, for that is a place with more robust information, currently.

for completeness (prompted by opetopic type theory) I started an entry *opetopic omega-category*.

For me presently this just serves to purpose to record Thorsten Palm’s definition of opetopic omega-category, as I understand it from what Eric Finster tells me.

For the definitions by Baez-Dolan and by Makkai the entry presently only contains placeholders, please feel invited to fill in detail.

All these definitions consider opetopic sets. The difference is in which structure and property is put on that. The original definition of universal cells is somewhat involved, as far as I see. Palm’s definition is of a nice straightforward homotopy-theoretic flavor. It seems plausible that this definition satisfies the homotopy hypothesis, but I don’t know if anyone looked into it.

Accoring to Eric Finster, Palm showed that his definition is a special case of Makkai’s, but the converse remains open.

]]>I have edited the old entry *n-fold category* a little, brought the content into proper order, fixed the link to cat-n-groups and cross-linked with *n-fold complete Segal space*.

I had set out to add to the entry *equivalence in homotopy type theory* a detailed derivation of the categorical semantics of $Equiv(X,Y)$. But then I ended up getting distracted by various editorial work in other entries and for the moment I only have this puny remark added, expanding on the previous discussion there.

Maybe more later…

]]>added to the references-section of the stub *type-theoretic model category* pointers to André Joyal’s slides on “typoses” (he is currently speaking about that again at CRM in Barcelona).

(maybe that entry should be renamed to “categorical semantics for homotopy type theory” or the like, but I won’t further play with it for the time being).

I am also pointing to Mike’s article and to his course notes. I will maybe ask André later, but I am a bit confused about (was already in Halifax) how he presents his typoses, without mentioning of at least very similar categorical semantics that has been discussed before. Maybe I am missing some sociological subtleties here.

]]>started bracket type, just for completeness, but don’t really have time for it

]]>I fixed some of the references at *Batanin omega-category*.

the graphics at the old entry *horizontal composition* comes out wrong on my system. What’s going on? This is included as SVG.

gave *Lagrangian cobordism* an Idea-section added references related to the Fukaya category and cross-linked with relevant entries.

added a few more references with brief comments to *QFT with defects*

(this entry is still just a stub)

]]>I am beginning to give the entry *FQFT* a comprehensive *Exposition and Introduction* section.

So far I have filled some genuine content into the first subsection *Quantum mechanics in Schrödinger picture*.

But I have to quit now. This isn’t even proof-read yet. So don’t look at it unless you feel more in editing-mood than in pure-reading-mood.

]]>at *braided monoidal 2-categiry* the following query box was sitting, which I hereby move from there to here

+–{: .query} Ben Webster: I would very much like to know: what structure on a (triangulated/dg-/stable infinity/whatever you like) monoidal category would make its 2-category of module categories (give that phrase any sensible construal you like) is braided monoidal.

If one decategorifies this question, one gets the question “what structure on a ring makes its category of representations braided monoidal” and the answer to this question is well-known: a quasi-triangular quasi-Hopf structure.

I asked a MathOverflow question on the same topic. No interesting answers yet. =–

]]>I have edited at *HQFT*, touched the general formatting and structuring a good bit, trying to clean it up and beautify it a bit, and added a brief cross-pointer to the cobordisms hypothesis for cobordisms with maps into a base manifold.

I have added to all *Segal space*-related entries, as well as to the Example section at *category object in an (infinity,1)-category* statements like

a pre-category object in $\infty Grpd$ is called a Segal space;

a connected pre-category object in $\infty Grpd$ is called a reduced Segal space;

a category object in $\infty Grpd$ is called a complete Segal space.

an category object in $Cat(Cat(\cdots Cat(\infty Grpd)))$ is called an n-fold complete Segal space;

That list can be further expanded. But I have to quit now.

]]>added some basics to *model structure for quasi-categories* at *general properties*

created *dg-nerve*

following Mike’s suggestion, I have split off from cohesive topos the entry

I used that splitting-off to *play Bourbaki* and decide that I don’t follow Lawvere’s definition in all detail. Instead, it seems to me we can usefully streamline it. It should say just two things: a cohesive $(\infty,1)$-topos is

locally and globally $\infty$-connected

local .

And that’s it. That gives the quadruple of adjoint functors, where the inverse image and its parallel functor are both full and faithful.

I have also added an Interpretation-section where I highlight that this implies two central properties of cohesive $(\infty,1)$-toposes:

they have the

*shape*of the point in the sense of shape of an (infinity,1)-topos (this is implied by local plus global $\infty$-connectedness);they look like small neighbourhoods of the standard point (this is what the locality axioms means, given the standard examples for local toposes).

added to *n-excisive functor* a section

Jamie Vicary is kindly adding information to the $n$Lab on the higher-category-theory-proof-assistant that he and collaborators are developing, at:

I have added a few more hyperlinks to related nLab entries.

And I have changed the page name from lower case “globular” to upper case “Globular” to fit our conventions on entry titles.

Currently, lower case “globular” still redirects to the entry. But if anyone has links to the lower case version from elsewhere, please consider changing them, for eventually the lower case “globular” really ought to go to a page that disambiguates all sorts of globular-related entries on the nLab, such as *globe* and *globular set*, etc.

added at cobordism hypothesis a pointer to

- Yonatan Harpaz,
*The Cobordism Hypothesis in Dimension 1*(arXiv:1210.0229)

where the case for $(\infty,1)$-categories is spelled out and proven in detail.

]]>I’d be trying to write out a more detailed exposition of how fiber integration in twisted generalized cohomology/twisted Umkehr maps are axiomaized in linear homotopy-type theory.

To start with I produced a dictionary table, for inclusion in relevant entries:

]]>In the process of expaning on [[n-truncated object in an (infinity,1)-topos]] I added some remarks along these lines to the beginning of [[homotopy n-type]], thereby rewriting the first few sentences.

]]>Created *opetopic type theory* with a bit of explanation based on what I understood based on what Eric Finster explained and demonstrated to me today.

This is the most remarkable thing.

I have added pointers to his talk slides and to his online opetopic type system, but I am afraid unguided exposition to either does not reveal at all the utmost profoundness of what Eric made me see when he explained and demonstrated OTT to me on his notebook. I hope he finds time and a way to communicate this insight.

]]>Somebody over lunch at the conference here said that the $n$-Lab somewhere leaves out a condition in the definition of n-fold complete Segal spaces, namely “it’s not just completeness, there is also a condition that many spaces are degenerate”.

We were offline and couldn’t quite determine which entry was meant. Now I am online but alone, and I checked at *n-fold complete Segal space*, which doesn’t really give any definition at all, but points to *(infinity,n)-category* and *n-category object in an (infinity,1)-category*. I *think* (am pretty sure) that there the correct definition is given, but I don’t really have the leisure to check in detail right now.

Instead, I suspect that everything on the nLab is correct but there is just a subtlety that maybe deserves to highligted more, namely for $n$-fold Segal spaces the completenss condition automatically involves more and more degeneracy condition due to the way that $\infty$-groupoids are regarded as degenerate cases of $(n-1)$-fold complete Segal spaces.

To hint at that (don’t have time for more right now), I have now added to *n-fold complete Segal space* the following paragraph:

]]>In analogy of how it works for complete Segal spaces, the completness condition on an $n$-fold complete Segal space demands that the $(n-1)$-fold complete Segal space in degree zero is (under suitable identifications) the infinity-groupoid which is the core of the (infinity,n)-category which is being presented. Since the embedding of $\infty$-groupoids into ($n-1$)-fold complete Segal spaces is by adding lots of degeneracies, this means that the completeness condition on an $n$-fold complete Segal space involves lots of degeneracy conditions in degree 0.

The reference provided today on the CatTheory mailing list

- A.R. Garzón, J.G. Miranda,
*Serre homotopy theory in subcategories of simplicial groups*Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123

I have added to k-tuply groupal n-groupoid, and also to n-group and infinity-group

]]>