Low-dimensional higher geometry by examples

In late 2016 I visited the University of Colorado, Boulder to talk with Carla Farsi, and I gave a colloquium talk there, entitled “Low-dimensional higher geometry by examples”. It used the blackboard, so here’s a scan of my notes.

A case of induction

I have finally proved the induction step for the construction of a Fréchet manifold that is a limit of a large diagram of such manifolds, by a very carefully chosen iterated sequence of pullbacks of submersions. The base case requires one to construct by hand, in the notation of the picture below, the manifolds $(I)$ for $|I|=1,2,3$. Here $I$ is a finite set, with elements $i,j,\ldots$, indexing the sets in a closed cover $\{V_i\}_{i\in I}$ of a compact manifold $M$, and the subsets $J=I\setminus i$ etc index ‘partial subcovers’. That is, the closed sets corresponding to the elements of that subset, which do not themselves form a cover. Let $V_J := \coprod_{j\in J} V_j$. The manifolds $(J)$ are a priori the diffeological spaces of functors $\check{C}(V_J) \to X$, where $X$ is a fixed finite-dimensional Lie groupoid, and here $\check{C}(V_J)$ is the diffeological Cech groupoid of $V_J \to M$. The aim here is to show that $(I) = \mathrm{Hom}(\check{C}(V_I),X)$ is in fact a Fréchet manifold, by induction on the size of $I$. This result is Proposition 5 of my MATRIX Annals note with Raymond Vozzo, The smooth Hom-stack of an orbifold (publisher link, arXiv), with the proof there only stating

This diffeological space is what we show is a Fréchet manifold, by carefully writing the limit as an iterated pullback of diagrams involving maps that are guaranteed to be submersions by Proposition 3 and Theorem 4, and using the fact that X is appropriately coskeletal

“Carefully writing the limit” indeed! Past me was highly constrained by page limits…

Localisation by cospans redux

In my previous post I described a construction whereby a 2-category of groupoids $Gpd_J^{ccof}(S)$ internal to a lextensive category $S$ whose 1-arrows are functors with object component a coproduct inclusion can be localised at the fully faithful essentially $\amalg J$-surjective functors. The 1-arrows of $Gpd_J^{ccof}(S)$ are called complemented cofibrations, and the 1-arrows of the localisation $Gpd_J^{ccof}(S)[W^{-1}]$ were then certain cospans of such functors. This was rather formal and relied on these notes of mine. However, for the case that the ambient category is that of sets, Land, Nikolaus and Szumiło showed that this localisation is equivalent to localising the category of all small groupoids at the fully faithful essentially surjective functors (although they worked in the setting of $(\infty,1)$-categories, the result is still just a $(2,1)$-category). I started sketching how one should construct a 2-functor $Gpd_J(S)\to Gpd_J^{ccof}(S)[W^{-1}]$ from a 2-category $Gpd_J(S)$ of internal groupoids with more general functors to the localisation by cospans. (The functors need to satisfy a condition related to the pretopology one is using on the lextensive category, which in the case of a pretopos with the canonical topology turns out to be always true, but there are interesting cases where it is a real restriction.) That said, one can use Pronk’s comparison theorem to show the stronger result that $Gpd_J(S)[W^{-1}]\simeq Gpd_J^{ccof}(S)[W^{-1}]$, which I will sketch below the fold.

Cofibrations of internal groupoids and localisations by cospans

In my paper ICAL, I dealt extensively with internal categories and groupoids, and how to localise 2-categories of these at the fully faithful essentially ‘surjective’ functors (‘surjective’ in the sense there is some well-behaved class of regular epimorphisms playing the rôle of surjective maps). Even for 1-categories of ordinary small groupoids or categories one can consider this as a variant on the canonical model structure on groupoids or on categories, and formally invert the fully faithful, essentially surjective functors, which aren’t invertible in the 1-category sense. Actually it really only uses the category of fibrant objects structure, and ICAL (and more in generality, these notes) can be seen as a way to make this work in a way that doesn’t use the full category of fibrant objects structure, but the 2-category structure instead. There is another way to think about this process for the case of small groupoids (and more generally from some cofibration category), which I learned from a paper of Land, Nikolaus and Szumiło, Localization of Cofibration Categories and Groupoid C*-algebras (publisher, arXiv) (cofibrations of groupoids are functors that are injective on objects). I want to outline how one can repeat the 2-categorical, extra weak story for internal groupoids (at least; I haven’t thought about the case of internal categories) using this type of approach. In some respects, I can get more general results, in that I don’t require that I’m localising a small cofibration category with good cylinders, but also kinda less general in the sense that the universal property I get is only for bicategories, not more general $\infty$-categories (presented by something Quillen equivalent to a combinatorial model category). Below the fold I set up some basic assumptions, and then outline the constructions that will go into the hypotheses of the construction.