**Changes-note**. Changed the already existing page 201707071634 to now contain a different svg illustration, planned to be used in an integrated way in pasting schemes soon.

**Metadata.** Like here, except that in 201707071634 symbols (arrows) indicating what is to be interpreted to 2-cells are given, in the same direction as in Power’s paper.

**Changes-note**. Changed the already existing page 201707071626 to now contain a different svg illustration, planned to be used in pasting schemes soon.

**Metadata.** What 201707051600 is: relevant material to create an nLab article on pasting schemes.
This is (a *labelling* of) the (plane diagram underlying the) pasting diagram A. J. Power gives as an example in his proof of his pasting theorem herein.

Unlike there, the 2-cells are not indicated in 201707051600.

Related concepts: pasting diagram, pasting scheme, digraph, planar graph, higher category theory.

]]>Changed 201707051620.

**Metadata.** by-and-large, cf. this thread.
A difference to 201707051600 is that here what A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990) calls a “boundary of the face F” is indicated by bold arrows.

EDIT: (proof of necessity of hypothesis in [A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990] and relevance to 201707051600 temporarily removed, to make it more uniform)

]]>EDIT:

**Changes note.** Changed the already existing page 201707040601 to contain an svg illustration relevant to pasting scheme and [this thread]

**Meta data.** cf. [this thread]; difference is that in 201707040601 a *face* $F$ of the plane digraph is named and one of the two orientations of the euclidean plane is indicated by a circular gray arrows. A connection to [Power’s proof] can be seen by letting $q_{-\infty}:=s$ (in Power’s sense), and $q_{\infty}=t$, and $F$ the “F” in Power’s paper.

OLD, bug-related discussion:

For some reason unknown to me, the “discussion” (actually, it is merely meant to be the obligatory “log what you do” entry), the discussion with name ‘201707040601’ that I started seems to have technical problems: the comment I entered is not displayed (to me). I would delete it, but apparently it is not possible to delete “discussions” one has started. Please do with it whatever seems most appropriate.

]]>**Changes-note**. Changed the already existing page 201707051600 I created, to now contain another svg illustration, planned to be used in pasting schemes soon. Sort-of-a-permission for this is

Power’s proof of (I guess you mean) his pasting theorem would probably be very handy to have discussed at the nLab. It would seem to fit at one of pasting diagram or pasting scheme, but less well at an article on some notion of graph I think. If you could even just write down the precise definitions of these various notions, that would also be very fine in my opinion.

**End of changes-note**

**Metadata.** What 201707051600 is: relevant material to create an nLab article on pasting schemes.
More specifically: to document A. J. Power’s proof of one of the rigorous formalizations of the notational practice of pasting diagrams.
201707051600 shows a plane digraph $G$.
Vertex $q_{-\infty}$ is an $\infty$-coking in $G$.
Vertex $q_{\infty}$ is an $\infty$-king in $G$.
Connection to A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990): therein, the author calls $q_{-\infty}$ a “source”, and $q_{\infty}$ a “sink”. This is fine but not in tune with contemporary (digraph-theoretic) terminology, whereas “king” and “coking” are.
These technical digraph-theoretic terms will be defined in digraph.

Related concepts: pasting diagram, pasting scheme, digraph, planar graph, higher category theory.

[ Some additional explanation: it was bad practice of me, partly excusable by the apparent LatestChanges-thread-starting-with-a-numeral-make-that-thread-invisible-forum-software-bug, to have created this page without notification and having it left unused for so long. Within reason, *every* illustration one publishes should be taken seriously, and documented. Much can be read on this of course, one useful reference for mathematicians is the TikZ&PGF manual, Version 3.0.0, Chapter 7, Guidelines on Graphics. My intentions were well-meant, in particular to improve the documentation of monoidal-enriched bicategories on the nLab. This is still work in progress, but to get the digraph/pasting scheme project under way is more urgent. Will re-use the 201707* named pages for this purpose, for tidiness. ]

[Reason for new thread: to all appearances, tricategory did not have one of its own, despite tetracategory having one]

(Updated reference to a representability theorem in arXiv:0711.1761v2 on tricategory; what was Theorem 21 in arXiv:0711.1761v1 has become Theorem 24 in arXiv:0711.1761v2 and its journal version)

]]>It would not surprise me if there already is a thread for this, but none was found: this thread is meant to collect (useful!, serious!, sparingly used!, carefully chosen!) verbal mnemonics/slogans/unusual-yet-useful-technical-terms relevant to category theory (especially higher-category theory). Some of them can also be a bit cryptic, and given without much explanations, such as the example below.

The emphasis should be more on compressed mnemonics than on sweeping slogans (nice though those can be).

This is not to get into a discussion about whether mnemonics are good or bad or tasteless.

I, too, am of the opinion that, by and large. structural thinking and “abstrakte Anschauung” is to be preferred to grade-school-like verbal crutches. In Dutch and the German the vernacular term for “mnemonic” is even inherently disparaging, by the way. And then again, mnemonics is sort of an art, with a venerable tradition since antiquity.

Here is an example, perhaps not even a particularly nifty one, to get this thread started:

(monoidal monoidoid)$=$(monobject bimonoidoid)

]]>Is the category Hom of bicategories with homomorphisms as the morphisms, in the sense of

Ross Street, Fibrations in bicategories, Cahiers de topologie et géométrie différentielle catégoriques, tome 21, no 2 (1980), p. 111-160

already (recognizably) documented on the nLab ? (I did a sem-cursory search in this respect, but did not find it documented (in its own right, I mean, the article of Street appears.)

Should it be?

Should it have an article of its own?

To me it seems it should (my motivation is that I am using and documenting bicategories currently, and are studying Street’s 1980 paper as a sort of background reading to Garner–Shulman, Adv. Math. 289), but its traditional name Hom seems unfortunate, creating yet another meaning of Hom.

My suggestion would be to call it (and its article)

$BiCat$

]]>With Urs Schreiber and Alessandro Valentino we are finalizing a short note on central extensions of mapping class groups from characteristic classes.

A preview of the note is available here: *Higher extensions of diffeomorphism groups (schreiber)*

Any comment or criticism is most welcome

]]>Hey all,

So I’ve been kind of bugging out trying to find some kind of coherent theory of comonoids in $\infty$-categories. This, for instance, would apply to comonads (as co-associative comonoids in endomorphism categories) among other things. When I try to use Lurie’s stuff, I end up having to trace further and further back to try to prove anything, and end up feeling like I need a theory of cooperads. Somehow comonoid structures seem fundamentally different than monoid structures. Does anyone know how to do this, or if it’s written down clearly anywhere? For instance, Lurie has this nice theorem in one of the DAGs where he shows that monoids are essentially simplicial objects, and this seems to generalize pieces of Emily Riehl’s work with Dominic Verity, except for the fact that there’s no analog for comonoidal objects. It’d be nice to have the analogous statement saying that comonoids are cosimplicial objects in some essential way.

Thanks for any ideas!

-Jon

]]>Hey again everybody,

So last night I was trying to prove that if we take the (∞,1)-category of (∞,1)-categories use the left adjoint to the inclusion of the n-truncated objects, we get a localization at a reasonable subcategory of things that we could call a quasicategory of (n,1)-categories or n-categories. I’m aware that all I can really hope to get is a category of objects which are equivalent to honest n-categories. Is this a reasonable thing to try to do? It seems like it would make proving things about n-categories, for instance, a monadicity theorem, easier. Perhaps this is well known, and I just haven’t searched well enough. Do you guys know?

-Jon

]]>So I’m trying to draw some kind of connections here between a lot of really useful stuff that’s written on the nlab. I think it might even deserve its own page, which I would title “Descent Cohomology” (although maybe this is actually really quite trivial and doesn’t need its own page) but I think I need some help from you all to make it make sense in the “$\infty$” case. A lot of this has kind of been inspired by reading “Principal $\infty$-bundles - General Theory” by Urs, Thomas Nikolaus and Danny Stevenson, as well as stuff by Lurie, and a ton of other stuff for the discrete (and 2-categorical) case (Nuss and Wambst, Larry Breen, Knus and Ojanguren, SGA4, etc).

Given a cover in some ($\leq\infty$-) site $C$, $\phi:U\to X$, and some stack (or categorical bifibration?) $\mathcal{F}:C\to \infty-Cat$, I’d like to answer the question, for $M\in \mathcal{F}(X)$, what other $N\in\mathcal{F}(X)$ are there (up to equivalence) such that $\phi^\ast(M)\simeq \phi^\ast(N)$. This is basically asking for “twisted forms” of $M$, and if I’m not mistaken, this should be, at least theoretically, calculable as some kind of “cohomology” in some $\infty$-topos.

In the discrete case, one can compute such a thing by looking at $\check{H}^1(U\overset{\phi}\to X,Aut(\phi^\ast(M)))$, I believe. However, this also seems to compute principal $Aut(M)$-bundles for that cover as well. And in the nice case that the cover is “Galois” for some group $G$, this can be written down in terms of group cohomology of $G$ with coefficients in $Aut(M)$. There is machinery for doing something similar even in the more general scenario of Hopf-Galois extensions by some Hopf-algebra, explained very nicely in the paper by Nuss and Wambst: Non-Abelian Hopf Cohomology. What’s really nice about that scenario is that this same cohomology also classifies descent data for $\phi^\ast(M)$ (continuing with the notation from above). That is, it also classifies isomorphisms between the two different ways to pull back $\phi^\ast(M)$ to $U\times_ U$ (excuse me for skimping on the explanation here, the notation just gets unpleasant), or if we’re in the situation of monadic descent for some monad $T$, it classifies comodule-structures on $\phi^\ast(M)$ for the relevant comonad on the category of $T$-algebras (a relatively nice account of this is given in Mesablishvili’s On Descent Cohomology as well as Menini and Stefan’s Descent Theory and Amitsur Cohomology of Triples . So, this one single cohomology group computes a whole bunch of different things, which are all actually the same thing, and if we have a nice enough cover, we have even nicer ways of computing it.

So I guess my question is the following - Given all that we know about $\infty$-principal bundles (being computed by some $\mathbf{H}(X,BG)$), can we recognize this as some kind of descent cohomology, or higher Amitsur cohomology in the case of descent for either $\infty$-stacks or derived stacks? Now, a descent datum should be, instead of an isomorphism with a cocycle condition, a isomorphism with all higher cocycle conditions and bunch of cells gluing all of this stuff together (or in other words, the category of descent data is the limit of some simplicial $\infty$-category (or colimit and cosimplicial, depending on variance and so forth)). And so the “descent cohomology” in this scenario should be some higher, or derived, mapping space, but it should also depend on the choice of cover.

I’m trying to pick up the $\infty$ -categorical pieces as fast as I can, but I was just wondering if anyone has thought about this particular situation. I’d really really love to chat about it and try to get it ironed out.

Thanks!

]]>have created *affine modality*

I have noticed the following error in the categorification of a local homeomorphism of spaces into a concept of a local homeomorphism of <latex>n</latex>-topoi. I have a suspicion that it must have been noticed before (at least for <latex>1</latex>-topoi), and if this is true, please view this discussion, in particular, as a reference request.

**Background**

For a topological space <latex>X,</latex> if <latex>F</latex> is a sheaf, it has an \‘etal\‘e space <latex>Y_F \to X</latex> which is a local homeomorphism over <latex>X,</latex> and sections of it are exactly the sheaf <latex>F.</latex> The \‘etal\‘e space construction yields an equivalence of categories

<latex>Sh\left(X\right) \simeq Et/X</latex> between the category of sheaves on <latex>X</latex> and the category of local homeomorphisms over <latex>X.</latex> Moreover, assuming <latex>X</latex> is sober, since sober topological spaces embed fully faithfully into topoi, for a sheaf <latex>F,</latex> <latex>Y_F \to X</latex> corresponds to a geometric morphism <latex>Sh(Y_F) \to Sh(X)</latex> and there turns out to be an equivalence of topoi <latex>Sh(Y_F) \simeq Sh(X)/F</latex> under which this geometric morphism is equivalent to the one induced by slicing: <latex>Sh(X)/F \to Sh(X).</latex>

Using this as an example, topos theorists said that a geometric morphism of <latex>1</latex>-topoi <latex>\mathcal{F} \to \mathcal{E}</latex> is **\‘etale** if it is equivalent to one of the form <latex>\mathcal{E}/E \to \mathcal{E},</latex> and this was to be a ``local homeomorphism’‘ of <latex>1</latex>-topoi.

**The Problem**

Define an \‘etale geometric morphism of <latex>n</latex>-topoi the same way, i.e.

<latex>\mathcal{F} \to \mathcal{E}</latex> is **\‘etale** if it is equivalent to one of the form <latex>\mathcal{E}/E \to \mathcal{E}.</latex>

Now let <latex>X</latex> be a (sober) topological space. In particular, it is a locale, hence a <latex>0</latex>-topos in the sense of Lurie. Lets write this <latex>0</latex>-topos by <latex>O(X)</latex> (as it is in fact the lattice of opens of <latex>X.</latex>) The maps of locales that are induced by slicing are exactly those which correspond to inclusions of open sublocales. So, an \‘etale map of <latex>0</latex>-topoi does not correspond to a local homeomorphism (but it is a particular kind of local homeomorphism).

You might ask: So what? Maybe this is just a defect for <latex>n=0.</latex>

No, this persists. An <latex>\infty</latex>-topos <latex>\mathcal{X}</latex> is **<latex>n</latex>-localic** if it is equivalent to <latex>\infty</latex>-sheaves on an <latex>n</latex>-site. There is a functor from <latex>n</latex>-topoi to <latex>\infty</latex>-topoi which is fully faithful and whose essential image is <latex>n</latex>-localic <latex>\infty</latex>-topoi, and if <latex>\mathcal{X}</latex> is an <latex>n</latex>-localic, the <latex>n</latex>-topos which is corresponds to is the <latex>\left(n,1\right)</latex>-category of <latex>\left(n-1\right)</latex>-truncated objects of <latex>\mathcal{X}.</latex>

Moreover, if <latex>X</latex> is an object of an <latex>\mathcal{X},</latex> then <latex>\mathcal{X}/X</latex> is also <latex>n</latex>-localic **if and only if** <latex>X</latex> is <latex>n</latex>-truncated. So let <latex>X</latex> be an object of <latex>\mathcal{X}</latex> which is <latex>n</latex>-truncated but not <latex>\left(n-1\right)</latex>-truncated. Then the \‘etale map of <latex>\infty</latex>-topoi <latex>\mathcal{X}/X \to \mathcal{X}</latex> is a morphism between <latex>n</latex>-localic <latex>\infty</latex>-topoi. Let <latex>\mathcal{F}</latex> denote the <latex>n</latex>-topos associated to <latex>\mathcal{X}/X</latex> and <latex>\mathcal{E}</latex> the one associated to <latex>\mathcal{X}.</latex> Since <latex>n</latex>-topoi embed fully faithfully into <latex>\infty</latex>-topoi as <latex>n</latex>-localic <latex>\infty</latex>-topoi, this \‘etale map induced by <latex>X</latex>

must correspond to a geometric morphism <latex>\mathcal{F} \to \mathcal{E}.</latex> However, it cannot be an \‘etale map of <latex>n</latex>-topoi, since the object <latex>X</latex> is not <latex>\left(n-1\right)</latex>-truncated. Nonetheless, such a geometric morphism is the “correct” notion of a local homeomorphism since if we let <latex>n=0</latex>, this is a local homeomorphism of locales, in the usual sense.

Any comments? Has this been noticed for <latex>1</latex>-topoi and is there a name for such geometric morphisms (and is there a characterization of them internal to <latex>1</latex>-topoi?)

]]>together ideas from "network theory" and "(higher) category theory" that

I don't know where to start. It would be useful if there was a context

specified (e.g. TQFTs).

Robert

Is there stuff on the nlab or mathoverflow or...

that my search abilities are not up to finding?

jim ]]>

I have added some discussion to the page on orientals (in the sense of Ross Street), regarding the link to the convex geometry of cyclic polytopes (as discussed by Kapranov and Voevodsky).

My selfish motive for doing so is that I am curious if my recent work with Steffen Oppermann which includes a new description of the triangulations of (even-dimensional) cyclic polytopes, has any relevance to the study of orientals, or higher category theory more broadly. (In particular, if there are explicit questions about the internal structure of orientals which are of interest, I would like to hear about them.)

A particularly speculative version of my question, would be whether there is a natural connection between orientals and the representation theory which we are studying in that paper (which necessitated a detour into convex geometry). We biject triangulations of an even-dimensional cyclic polytope to (a nice class of) tilting objects for a certain algebra. The simplest version of this (which was already known) is that triangulations of an $n$-gon correspond to tilting objects for the path algebra of the quiver consisting of a directed path with $n-2$ vertices. (These tilting objects then give derived equivalences between the derived category of this path algebra, and the derived category of the endomorphism ring of this tilting object.)

Questions, speculations, or suggestions would be very welcome.

Hugh

]]>Lemma: Any left anodyne map in SSet/S is a covariant equivalence.

Proof: We can consider the case of a left horn inclusion, since these generate all left anodyne maps.

Then we must show that any map

$i: LeftCone(\Lambda^n_j) \coprod_{\Lambda^n_j} S \to LeftCone(\Delta^n) \coprod_{\Delta^n} S$is a categorical equivalence. **However, $i$ is a pushout of of the map $\Lambda^{n+1}_{j+1} \to \Delta^{n+1}$, which is inner anodyne, so we’re done.**

Questions:

How do we show that $i$ is a pushout as described in the bolded sentence?

]]>