Timeline for combinatorical description of classifying map for principal $G$-bundle over Delta set
Current License: CC BY-SA 4.0
18 events
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| Feb 12 at 3:49 | comment | added | Dmitri Pavlov | @JackYo: Some formulas and definitions for the shape functor can be found here: ncatlab.org/nlab/show/shape+via+cohesive+path+∞-groupoid | |
| Feb 12 at 0:07 | comment | added | JackYo | Could you make precise what you mean in this context that object $A$ "is shape of" object $B$ (as you remarked in previous comment on relation between $BG^S$ and presheaf $BG$)? Similarly, that something is equivalent "to the shape" of another object? Does the term "shape" in this context have a precise meaning? | |
| Feb 11 at 23:24 | comment | added | Dmitri Pavlov | @JackYo: Maps into BG^S and BG classify different things. The derived space of maps of spaces X→BG^S is equivalent to the ∞-groupoid of principal G-bundles over X, their concordances, and higher concordances (parametrized by the simplex Δ^k) as k-morphisms. The derived space of maps RMap(X,BG) of ∞-sheaves (and not just spaces) X→BG is equivalent to the groupoid of principal G-bundles over X and their isomorphisms. Finally, the ∞-groupoid RMap(X,BG^S) is equivalent to the shape of the derived internal hom RHom(X,BG), whereas RMap(X,BG) is equivalent to RHom(X,BG)(R^0). | |
| Feb 11 at 21:00 | comment | added | JackYo | but because of this remark that $BG^S$ has rather the shape of the presheaf $BG$, $\text{RMap}(X,BG^S)$ inherits groupoid structure, so a kind of replacement argument. That's the argument you use there, right? But could you lose some words what you formally precisely mean by that "$BG^S$ has rather the shape of the presheaf $BG$"? It that a kind of "higher homotopy equivalence" allowing replacements preserving properties, as eg here for "beeing groupoid"? | |
| Feb 11 at 20:52 | comment | added | JackYo | What do you precisely mean by that the simplicial set you constructed in first two paragraphs - a model of classifying space, let call it $BG^S$ in order distinguish it from "the other, say "canonical" presheaf $BG$" - is rather the shape of the the latter $BG$? I assume that the way you are going to use it next, is that generally a derived mapping object with target beeing a groupoid carries naturally groupoid structure, so $\text{RMap}(X,BG)$ would have groupoid structure "for free" for $BG$ the "canonical" presheaf of groupoids, | |
| Feb 11 at 18:47 | comment | added | Dmitri Pavlov | @JackYo: Spaces could be taken to mean simplicial sets here. Groupoids embed into simplicial sets via the nerve functor. BG is a presheaf of groupoids (and therefore simplicial sets) on the site of cartesian spaces. (The space BG used in the first two paragraphs of my answer is not the same BG, but rather the shape of the presheaf BG.) Thus, the derived mapping object from X to BG is itself naturally a groupoid, the groupoid of principal G-bundles over X. | |
| Feb 11 at 13:47 | comment | added | JackYo | One little nitpick, you suggested in the brackets that the dervied mapping space $\text{RMap}(X,BG)$ carries naturally even a structure of a groupoid. Do I understand it correctly that in this framework your all "spaces" are considered automatically as "groupoids", so even when we start naively from a priori "just a space" - like $\text{RMap}(X,BG)$ - you implicitly treat it in this context as a groupoid due to some natural embedding "in the background" of these "spaces" in the bigger category of groupoids, right? In other words in spirit of the slogan "spaces are groupoids"? | |
| Feb 11 at 0:50 | vote | accept | JackYo | ||
| Feb 11 at 0:45 | comment | added | Dmitri Pavlov | @JackYo: Yes, that's correct. The groupoid of principal G-bundles over X is the dervied mapping space (or groupoid) RMap(X,BG). RMap can be computed as the right derived functor of Map (the simplicial mapping space), by cofibrantly replacing X and fibrantly replacing BG. If BG is an ∞-sheaf, then BG is fibrant in the local projective model structure on simplicial presheaves. A cofibrant resolution of X is given by the Čech nerve of a good open cover of X. Altogether, we get the groupoid of maps from the Čech nerve to BG. Thus, Čech cohomology computes sheaf cohomology in this context. | |
| Feb 10 at 23:05 | comment | added | JackYo | Furthermore, the last part I not completely understand: When we now know that the delooping $\infty$-presheaf is a actually a $\infty$-sheaf, why this implies that a descent datum for a principal $G$-bundle can be expressed as a morphism the Čech nerve of a good open cover to the delooping $BG$ as $\infty$-presheaf? Keeping this on more elementary level presheaf being sheaf means just that descent is effective. But how this "translates" to the last quoted statement with morphism from Cech nerve to $BG$? | |
| Feb 10 at 23:04 | comment | added | JackYo | If we now bulk up our objects to $\infty$-(pre)sheaves, do I understand it correctly that for a $\infty$-presheaf to be a sheaf is by definition equivalent to satisfy this "homotopy descent property" (ncatlab.org/nlab/show/…) where essentially lim is weakened to holim? | |
| Feb 10 at 23:03 | comment | added | JackYo | Thank you. I have to admit that I'm far from beeing familar with language of higher algebra, so maybe this question becomes nearly trivial: In your paper Prop 4.13 states that when a sheaf $G$ of groups on the site Cart, it is objectwise delooping $(M \to BG(M))$ - where I assume that there you use the term "objectwise" at that stage to ephasise that you consider it a priori as a presheaf - satisfies the homotopy descent property. | |
| Feb 8 at 19:18 | comment | added | Dmitri Pavlov | @JackYo: There is no equivalence relation used in the construction. I also added a paragraph about the other approach, where the answer is positive at least in some cases. | |
| Feb 8 at 19:17 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Feb 8 at 16:13 | comment | added | JackYo | What "leads" me to this conjecture that this "could maybe" work: Essentially the $BG$ for finite $G$ is a model for $1$-homotopy type, so it reasonable to expect that the neccessary data for construction of classifiyng map associated to principal $G$-bundle $p:P \to X$ over $X$ "simple enough" - here a finite Delta set - lying in $1$-skeleton might be enough to fully construct $f_p: X \to BG$, or is my intuition misleading me at that point? | |
| Feb 8 at 16:13 | comment | added | JackYo | Another point, pursuing the approach I sketched above from the "Here an approach..." part: Do you know if that works ...at least conceptually? So in certain sense if the "data" of a principal G-bundle $p:P \to X$ over a Delta set $X$ is only concentrated in the sense obove in dimension $1$ (ie from vertices & edges). And to construct the classifying map $f_p:X \to BG$ it suffice to determine it on $1$-skeleton level as a tried and then in this can (up to homotopy) it extends to $f_p$ whose pullback of universal bundle $EG \to BG$ is isom to to the origical $p$? | |
| Feb 8 at 16:09 | comment | added | JackYo | That's an interesting model, so to be sure, it's $n$-simplices $(\def\B{{\sf B}}\B G)_n$ are really literally formed by the set of principal $G$-bundles over standard $n$-simplex $\Delta^n$ and not(!) the set of isom classes of prin $G$-bdles over $\Delta^n$ (which would be a singleton as $\Delta^n$ contractible). In other words $(\def\B{{\sf B}}\B G)_n$ it distinguishes between "as naked objects" different, but isomorphic prin $G$-bdles from each other or do you pose in $(\def\B{{\sf B}}\B G)_n$ some equivalence relation (eg up to scaling) beeing weaker than isom relation? | |
| Feb 7 at 1:04 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |