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November 23, 2023

Classification of Metric Fibrations

Posted by Tom Leinster

Guest post by Yasuhiko Asao

In this blog post, I would like to introduce my recent work on metric fibrations following the preprints Magnitude and magnitude homology of filtered set enriched categories and Classification of metric fibrations.


First, we introduce the place we work, Fsetcat\mathsf{Fsetcat}, that is a category of categories enriched over the category of filtered sets (Fset\mathsf{Fset}). The symmetric monoidal category Fset\mathsf{Fset} contains Set\Set and [0,][0, \infty] as full submonoidal categories, hence the category Fsetcat\mathsf{Fsetcat} contains Cat\mathsf{Cat} and GMet\mathsf{GMet} (category of generalized metric space). Then by the framework of Leinster and Leinster–Shulman, we can consider the magnitude of “finite” Fset\mathsf{Fset}-categories, that is an extension of both of the Euler characteristic of finite categories and the magnitude of finite metric spaces, introduced by Leinster.

Second, we give a description of metric fibration, introduced by Leinster, as the “Grothendieck fibration”. In precise terms, we generalize the notion of Grothendieck fibration of small categories in Fset\mathsf{Fset}-enriched setting, and then we restrict it to Met\mathsf{Met} (category of metric spaces). Such a restriction coincides with the definition of metric fibrations. Further we obtain the notion of metric actions that plays a role of lax functors in Cat\mathsf{Cat} case, and we show a correspondence between them via the Grothendieck construction.

Finally, we give a classification of metric fibrations, which is parallel to that of topological fiber bundles. That is, the classification of metric fibrations is reduced to that of “principal fibrations” (torsors in Met\mathsf{Met}), which is done by the “1-Čech cohomology” in an appropriate sense. Further, we can define the “fundamental group π 1 m(X)\pi^m_1(X)” of a metric space XX, which is a group object in Met\mathsf{Met}, such that the conjugation classes of homomorphisms Hom(π 1 m(X),𝒢)\mathsf{Hom}(\pi^m_1(X), \mathcal{G}) corresponds to the isomorphism classes of 𝒢\mathcal{G}-torsors over XX. Namely, it is classified like topological covering spaces.

Filtered sets

First of all, we introduce a workspace Fsetcat\mathsf{Fsetcat} that contains Cat\mathsf{Cat} and GMet\mathsf{GMet}. This large category inspires us to deal with objects sometimes like a small category and sometimes like a metric space.


  • A filtered set is a set XX with subsets X XX_{\ell} \subset X for any [0,]\ell \in [0, \infty] satisfying that X X X_{\ell} \subset X_{\ell'} for any \ell \leq \ell' and X = 0X =XX_\infty = \bigcup_{\ell \in \mathbb{R}_{\geq 0}}X_{\ell} = X. We formally define that X =X_{\ell} = \emptyset for <0\ell \lt 0.

  • A filtered map f:XYf : X \longrightarrow Y} is a map with fX Y f X_{\ell} \subset Y_{\ell} for any [0,]\ell \in [0, \infty].

  • We define XYX \otimes Y by (XY) = +=X ×X (X\otimes Y)_{\ell} = \bigcup_{\ell' + \ell'' = \ell}X_{\ell'} \times X_{\ell''}.

  • For xXx \in X, we define degx=\deg x = \ell if xX <X x \in X_{\ell} \setminus \bigcup_{\ell' \lt \ell }X_{\ell'}. We have deg(x,y)=degx+degy\deg(x, y) = \deg x + \deg y for any (x,y)XY(x, y) \in X\otimes Y.

We denote the category of filtered sets by Fset\mathsf{Fset}. We can consider Fset\mathsf{Fset} as the subcategory of separated presheaves in Set [0,] op\Set^{[0, \infty]^{\mathrm{op}}}, and then the tensor product is the Day convolution.

We denote the yoneda functor [0,]Set [0,] op[0, \infty] \longrightarrow \Set^{[0, \infty]^{\mathrm{op}}} by y\mathsf{y}. The filtered set y(r)\mathsf{y}(r) consists of only one element with degree rr. Note that y()=\mathsf{y} (\infty) = \emptyset. It can be checked that (Fset,)(\mathsf{Fset}, \otimes ) is a symmetric monoidal category with the unit object *(0)\ast(0).

Further, there is a symmetric monoidal embedding SetFset\Set \longrightarrow \mathsf{Fset} sending a set XX to a filtered set with degx=0\deg x = 0 for any xXx \in X. Now we have the following left diagram of monoidal embedding, that induces the right diagram of embedding.

Here, we denote the category of Fset\mathsf{Fset}-categories by Fsetcat\mathsf{Fsetcat}. We also denote the category of [0,][0, \infty]-categories by GMet\mathsf{GMet}, the category of generalized metric spaces.

Benefits from Fsetcat\mathsf{Fsetcat}

Examples of magnitude

Before we consider metric fibrations from our view point Fsetcat\mathsf{Fsetcat}, we list up here some other benefits we obtain from there.

  • We can define magnitude and (co)weighting on a class of Fset\mathsf{Fset}-categories with a finiteness condition. It contains usual magnitude and (co)weighting of finite metric spaces, and also the Euler characteristic of finite categories introduced by Leinster.

  • We can consider magnitude (co)weighting for broader class of metric spaces than so far. (This is actually not a benefit from the consideration of Fsetcat\mathsf{Fsetcat} essentially, but I reached it in the process of defining magnitude on Fsetcat\mathsf{Fsetcat}. ) Namely, we can consider magnitude (co)weighting of locally finite graphs being not necessarily finite.

    If a metric space is finite, then we can compute its magnitude by summing up the magnitude (co)weighting. If a graph is not finite, it is not possible since the sum diverges, however, we can still consider its (co)weighting when the graph is locally finite.

    For example, let’s consider the Cayley graph of a finitely generated group. Then its magnitude (co)weighting takes the same value at each vertex, and it coincides with the inverse of growth series. The growth series of a finitely generated group is defined as gGq wlg\sum_{g \in G} q^{\mathsf{wl}g}. Here, wl\mathsf{wl} is the word length with respect to a generating system, and the “inverse” is taken in the formal power series ring [[q]]\mathbb{Z}[[q]]. This observation may be a foothold for applying magnitude theory to geometric group theory. We can also consider the Poincaré polynomial of a ranked poset as a magnitude weighting. (This is possible without extending the definitions of magnitude.)

MH as HH

We can also define magnitude homology of Fset\mathsf{Fset}-categories. Leinster–Shulman pointed out that the magnitude homology has a form of Hochschild homology in a generalized sense. Here, by considering a metric space like a small category in Fsetcat\mathsf{Fsetcat}, we obtain a more ring theoretic description.

Theorem   For CFsetcatC \in \mathsf{Fsetcat}, we have an isomorphism MH CGr HH (GrP C(),M C()) \mathsf{MH}^{\ell}_\bullet C \cong \mathrm{Gr}_\ell \mathsf{HH}_\bullet(\mathrm{Gr}P_C(\mathbb{Z}), M_C(\mathbb{Z})) for any 0\ell \in \mathbb{R}_{\geq 0}. Hence we have 0MH CHH (GrP C(),M C()). \bigoplus_{\ell\geq 0}\mathsf{MH}^{\ell}_\bullet C \cong \mathsf{HH}_\bullet({\mathrm{Gr}}P_C(\mathbb{Z}), M_C(\mathbb{Z})).

Here, GrP C()\mathrm{Gr}P_C(\mathbb{Z}) is the “category algebra of Fset\mathsf{Fset}-categories” defined as follows.

Definition   For CFsetcatC \in \mathsf{Fsetcat}, we define a filtered ring GrP C()\mathrm{Gr}P_C(\mathbb{Z}) by:

  • GrP C()=MorC\mathrm{Gr}P_C(\mathbb{Z}) = \mathbb{Z}\mathrm{Mor} C with (GrP C()) ={fMorCdegf}(\mathrm{Gr}P_C(\mathbb{Z}))_{\ell} = \mathbb{Z} \{f \in \mathrm{Mor} C \mid \deg f \leq \ell\} as a filtered abelian group.

  • For any f,gMorCf, g \in \mathrm{Mor} C, we define an associative product \cdot by
    fg={gf if  tf=sg  and  deggf=degf+degg, 0 otherwise. f\cdot g = \begin{cases}g\circ f & \text{if }&nbsp; t f = s g &nbsp;\text{ and }&nbsp; \deg g\circ f = \deg f + \deg g, \\ 0 & \text{ otherwise}.\end{cases}

We also define an action of GrP C()\mathrm{Gr}P_C(\mathbb{Z}) on the abelian group M C():=(ObC×ObC)M_C(\mathbb{Z}) := \mathbb{Z}(\mathrm{Ob} C\times \mathrm{Ob} C) from the right and the left by

f(a,b)={(sf,b) if  tf=a, 0 otherwise, f\cdot (a, b) = \begin{cases} (s f, b) & \text{ if }&nbsp; t f = a , \\ 0 & \text{ otherwise},\end{cases}


(a,b)f={(a,tf) if  sf=b, 0 otherwise.(a, b) \cdot f = \begin{cases} (a, t f) & \text{ if }&nbsp; s f = b , \\ 0 & \text{ otherwise}.\end{cases}

We use techniques of homology theory for small categories to give such an expression. This can also be considered as a kind of generalization of Gerstenhaber–Schack’s theorem asserting that the cohomology of a simplicial complex is isomorphic to the Hochschild homology of the incidence algebra.

Homotopies on Fsetcat\mathsf{Fsetcat}

By the usual nerve construction for small categories, we can construct a filtered simplicial set and filtered chain complex from a Fset\mathsf{Fset}-category. We say a filtered set XX is \mathbb{Z}-filtered if any xXx \in X has an integral degree. We denote the category of \mathbb{Z}-filtered sets by Fset\mathbb{Z}\mathsf{Fset}.

For a Fset\mathbb{Z}\mathsf{Fset}-category, we have a \mathbb{Z}-filtered chain complex by the above construction, and we can construct a spectral sequence EE from it. Then we have the following.

Theorem   Let CFsetcatC \in \mathbb{Z}\mathsf{Fsetcat}.

  • E p,q 1=MH p+q pCE^1_{p, q} = \mathsf{MH}^{p}_{p+q}C.

  • If CC is a digraph, we have E p,0 2=H˜ pCE^2_{p, 0} = \widetilde{H}_p C, where H˜ \widetilde{H}_\bullet denotes the reduced path homology introduced by Grigor’yan–Muranov–Lin–S.-T. Yau et al.

  • It converges to the homology of the underlying small category C̲\underline{C} if it converges, in particular when max{degffMorC}\max \{\deg f \mid f \in \mathrm{Mor} C\} is finite.

Now we discuss homotopy invariance of each page of this spectral sequence. We show that the (r+1)(r+1)-th page of this spectral sequence is invariant under “rr-homotopy”.

Two morphisms f,g:CDFsetcatf, g : C \longrightarrow D \in \mathsf{Fsetcat} are 11-step rr-homotopic if there is a morphism H:CI rDH : C \otimes I_r \longrightarrow D with H 0=f,H 1=gH_0 = f, H_1 = g. Here, I rI_r is the Fset\mathsf{Fset}-category with just two objects 0,10, 1 and one non-trivial morphism from 00 to 11 with degree rr.

The equivalence relation generated by “11-steps” is the rr-homotopy. For example:

  • 00-homotopy for small categories is exactly the natural transformation.

  • 11-homotopy for digraphs is exactly the digraph homotopy.

  • rr-homotopy for metric spaces is exactly the rr-closeness of Lipschitz maps. Namely, Lipschitz maps f,g:XYf, g : X \longrightarrow Y are rr-homotopic if and only if d(fx,gx)rd(f x, g x) \leq r for any xXx \in X.

Theorem   The (r+1)(r+1)-page of the above spectral sequence E r+1E^{r+1}_{\bullet\bullet} is invariant under rr-homotopy.

This contains the “digraph homotopy invariance of GLMY homology”. Further, this result suggests us that there are “rr-homotopy theories” on Fsetcat\mathsf{Fsetcat} that are the lifts of the following model structures on filtered chain complexes.

Theorem (J. Cirici, D. -E. Santander, M. Livernet and S. Whitehouse)  On the category of bounded filtered chain complexes, there is a model structure M rM_r for each r0r\geq 0 such that the weak equivalences are the chain maps inducing quasi-isomorphisms on the E rE^r term.

We can also expect that the following cofibration category structure on digraphs is a part of the “1-homotopy theory”.

Theorem (D. Carranza, B. Doherty, M. Opie, M. Sarazola and L.-Z. Wong)   On the category of digraphs, there is a cofibration category structure such that the weak equivalences are the digraph maps inducing isomorphisms on the GLMY homology.

Metric fibrations

Original definition by Leinster

First we recall the definition of metric fibration given by Leinster.

Definition (Leinster) Let π:EX\pi : E \longrightarrow X be a Lipschitz map between metric spaces. We say that π\pi is a metric fibration over XX if it satisfies the following: for any εE\varepsilon \in E and xXx \in X, there uniquely exists e xπ 1x\e_x \in \pi^{-1}x such that

  • d E(e,e x)=d X(πe,x)d_E(\e, \e_x) = d_X(\pi \e, x),

  • d E(e,e)=d E(e,e x)+d E(e x,e)d_E(\e, \e') = d_E(\e, \e_x) + d_E(\e_x, \e') for any eπ 1x\e' \in \pi^{-1}x.

We denote the category of metric fibrations over XX by Fib X\mathsf{Fib}_X. The morphisms are base and fiber preserving Lipschitz maps.

A remarkable property of the metric fibration is that the magnitude behaves like the Euler characteristic of topological fiber bundles as follows.

Proposition (Leinster) Let π:EX\pi : E \longrightarrow X be a metric fibration. If EE is a finite metric space, then its magnitude is a product of those of XX and π 1x\pi^{-1}x for any xXx \in X.

We note that fibers of π\pi are isometric.


  • For a product of metric spaces E=X×YE = X\times Y, the projection X×YXX\times Y \longrightarrow X is a metric fibration.

  • The projection from the complete bipartite graph K 3,3K_{3, 3} to K 3K_3 is a metric fibration. Hence the magnitude of K 3,3K_{3,3} is same as that of K 3×K 2K_3 \times K_2 as pointed out by Leinster.

The graph on the left is K 3×K 2K_3\times K_2, and the graph on the right is isomorphic to K 3,3K_{3, 3}. They both have the magnitude 61+3q+2q 2\frac{6}{1 + 3q + 2q^2}.

Metric fibrations as Grothendieck fibrations

As is well known to category theorists, there is a notion of fibrations for small categories called Grothendieck fibrations, of which we don’t explain the definition here. This notion is equivalent to lax functors via the so-called Grothendieck construction, and we can generalize this story in Fset\mathsf{Fset}-enriched setting. What is remarkable is that the restriction of such a generalized Grothendieck fibration to Met\mathsf{Met} (the category of metric spaces and Lipschitz maps) is exactly the metric fibration. We define the metric counterpart of lax functors as follows.

Definition  Let XX be a metric space.

  • A metric action F:XMetF : X \longrightarrow \mathsf{Met} consists of metric spaces FxMetF x \in \mathsf{Met} for any xXx \in X and isometries F xx:FxFxF_{x x'} : F x \longrightarrow F x' for any x,xXx, x' \in X satisfying the following for any x,x,xXx, x', x'' \in X:

    • F xx=id FxF_{x x} = {\mathrm{id}}_{F x} and F xx=F xx 1F_{x'x} = F_{x x'}^{-1},
    • d Fx(F xxF xxa,F xxa)d X(x,x)+d X(x,x)d X(x,x)d_{F x''}(F_{x'x''}F_{x x'}a, F_{x x''}a) \leq d_X(x, x') + d_X(x', x'') - d_X(x, x'') for any aFxa \in F x.
  • A metric transformation θ:FG\theta : F \Rightarrow G consists of Lipschitz maps θ x:FxGx\theta_x : F x \longrightarrow G x for any xXx \in X satisfying that G xxθ x=θ xF xxG_{x x'}\theta_x = \theta_{x'}F_{x x'} for any x,xXx, x' \in X. We can define the composite of metric transformations θ\theta and θ\theta' by (θθ) x=θ xθ x(\theta'\theta)_x = \theta'_x\theta_x. We denote the category of metric actions XMetX \longrightarrow \mathsf{Met} and metric transformations by Met X\mathsf{Met}_X.

The equivalence of Grothendieck fibrations and lax functors is generalized to Fset\mathsf{Fset}-enriched situation and restricted to Met\mathsf{Met} as follows.

Theorem   The Grothendieck construction gives a category equivalence Met XFib X. \mathsf{Met}_X \simeq \mathsf{Fib}_X.

Classification of metric fibrations

Now we give a complete classification of metric fibrations by several means, which is parallel to that of topological fiber bundles. Namely, we define “principal fibrations”, “fundamental groups” and a “11-Čech cohomology” for metric spaces, and obtain equivalences between categories of these objects. Roughly speaking, we obtain an analogy of the following correspondence in the case of topological fiber bundles with a discrete structure group.

Fiber bundles over XX with structure group GG

\leftrightarrow Principal GG-bundles over XX (GG-torsors)

\leftrightarrow [X,BG]Hom(π 1(X),G)/conjugation[X, B G] \cong \mathsf{Hom}(\pi_1(X), G)/conjugation

\leftrightarrow H 1(X,G)\mathsf{H}^1(X, G)

We explain these correspondences in the following, and we start from the definition of torsors in Met\mathsf{Met}.


We can define a subcategory Tors X 𝒢\mathsf{Tors}_X^{\mathcal{G}} of Fib X\mathsf{Fib}_X that consists of “principal 𝒢\mathcal{G}-fibrations” called 𝒢\mathcal{G}-torsors. Here and in the following, a group 𝒢\mathcal{G} is not just a group but is a group object of Met\mathsf{Met}, which we call a metric group. Namely, it is a metric space as well as a group whose operations are all Lipschitz maps.

Definition   Let GG be a group (not a metric one). A metric fibration π:EX\pi : E \longrightarrow X is a GG-torsor over XX if it satisfies the following:

  • GG acts isometrically on EE from the right, and preserves each fiber of π\pi.

  • Each fiber of π\pi is a right GG-torsor in the following sense. Let GG be a group and YY be a metric space. We say that YY is a right GG-torsor if GG acts on YY from the right and satisfies the following :

    • It is free and transitive.
    • g:YYg : Y \longrightarrow Y is an isometry for any gGg \in G.
    • We have d Y(y,yg)=d Y(y,yg)d_Y(y, yg) = d_Y(y', y'g) for any y,yYy, y' \in Y and gGg \in G.

For a GG-torsor π:EX\pi : E \longrightarrow X, we can equip GG with a metric group structure that is isometric to any fiber of π\pi. Hence, in the following, we write “𝒢\mathcal{G}-torsors” instead of “GG-torsors”, where 𝒢\mathcal{G} is such a metric group obtained from GG.

A 𝒢\mathcal{G}-morphism between 𝒢\mathcal{G}-torsors is a GG-equivariant map that is also a morphism of metric fibrations. We denote the category of 𝒢\mathcal{G}-torsors over XX and 𝒢\mathcal{G}-morphisms by Tors X 𝒢\mathsf{Tors}_X^{\mathcal{G}}. Note that the category Tors X 𝒢\mathsf{Tors}_X^{\mathcal{G}} is a subgroupoid of Fib X\mathsf{Fib}_X.

On the other hand, we can also define a subcategory PMet X 𝒢\mathsf{PMet}_X^{\mathcal{G}} of Met X 𝒢\mathsf{Met}_X^{\mathcal{G}} that is the counterpart of Tors X 𝒢\mathsf{Tors}_X^{\mathcal{G}}.

Definition   Let XX be a metric space and 𝒢\mathcal{G} be a metric group.

  • A 𝒢\mathcal{G}-metric action F:XMetF : X \longrightarrow \mathsf{Met} is a metric action satisfying the following:

    • F x=𝒢F_x = \mathcal{G} for any xXx \in X.
    • F xxF_{x x'} is a left multiplication by some f xx𝒢f_{x x'} \in \mathcal{G} for any x,xXx, x' \in X.
  • Let F,G:XMetF, G : X \longrightarrow \mathsf{Met} be 𝒢\mathcal{G}-metric actions. A 𝒢\mathcal{G}-metric transformation θ:FG\theta : F \Rightarrow G is a metric transformation such that each component θ x:FxGx\theta_x : F x \longrightarrow G x is a left multiplication by an element θ x𝒢\theta_x \in \mathcal{G}. We denote the category of 𝒢\mathcal{G}-metric actions XMetX \longrightarrow \mathsf{Met} and 𝒢\mathcal{G}-metric transformations by PMet X 𝒢\mathsf{PMet}_X^{\mathcal{G}}.

Apparently, PMet X 𝒢\mathsf{PMet}_X^{\mathcal{G}} is a subcategory of Met X\mathsf{Met}_X and is also a groupoid. Then we have the following.

Theorem   The Grothendieck construction gives a category equivalence PMet X 𝒢Tors X 𝒢. \mathsf{PMet}_X^\mathcal{G} \simeq \mathsf{Tors}_X^\mathcal{G}.

Fundamental metric group

As an example of a metric group, we construct the fundamental metric group π 1 m(X,x)\pi_1^m(X, x) of a metric space XX. Roughly speaking, it is a collection of “loops in XX” based at some xXx \in X, which are identified by a “homotopy relation”. In precise terms, we define it as follows.

Definition   Let XX be a metric space and xXx \in X.

  • For each n0n \geq 0, we define a set P n(X,x)P_n(X, x) by P n(X,x):={(x,x 1,,x n,x)X n+2}. P_n(X, x) := \{(x, x_1, \dots, x_n, x) \in X^{n+2}\}. We also define that P(X,x):= nP n(X,x)P(X, x) := \bigcup_n P_n(X, x).

  • We define a connected graph G(X,x)G(X, x) with the vertex set P(X,x)P(X, x) as follows. For u,vP(X,x)u, v \in P(X, x), an unordered pair {u,v}\{u, v\} spans an edge if and only if it satisfies both of the following :

    • There is an n0n \geq 0 such that uP n(X,x)u \in P_n(X, x) and vP n+1(X,x)v \in P_{n+1}(X, x).
    • There is a 0jn0 \leq j \leq n such that u i=v iu_i = v_i for 1ij1 \leq i \leq j and u i=v i+1u_i = v_{i+1} for j+1inj+1 \leq i \leq n, where we have u=(x,u 1,,u n,x)u = (x, u_1, \dots, u_n, x) and v=(x,v 1,,v n+1,x)v = (x, v_1, \dots, v_{n+1}, x).
  • We equip the graph G(X,x)G(X, x) with a weighted graph structure by defining a function w G(X,x)w_{G(X, x)} on edges by
    w G(X,x){u,v}={d X(v j,v j+1)+d X(v j+1,v j+2)d X(v j,v j+2) v jv j+2, 0 v j=v j+2, w_{G(X, x)}\{u, v\} = \begin{cases} d_X(v_j, v_{j+1}) + d_X(v_{j+1}, v_{j+2}) - d_X(v_j, v_{j+2}) & v_j\neq v_{j+2}, \\ 0 & v_j = v_{j+2}, \end{cases} where we use the notation v jv_j as in the previous bullet point.

  • We denote the quasi-metric space obtained from the weighted graph G(X,x)G(X, x) by Q(X,x)Q(X, x). Let π 1 m(X,x)\pi_1^{m}(X, x) be the metric space obtained from Q(X,x)Q(X, x) by identifying points with distance 00. The metric space π 1 m(X,x)\pi^m_1(X, x) has a metric group structure given by the concatenation. We call the metric group π 1 m(X,x)\pi_1^m(X, x) the fundamental metric group of XX with the base point xx.

We sometimes write π 1 m(X)\pi_1^m(X) since it does not depend on the choice of the base point, similarly to the topological case. We also define a category Hom(π 1 m(X),𝒢)\mathsf{Hom}(\pi_1^m(X), \mathcal{G}) of homomorphisms π 1 m(X)𝒢\pi_1^m(X) \longrightarrow \mathcal{G}, where a morphism between homomorphisms is a conjugation relation. Then we have the following.

Theorem   We have a category equivalence Hom(π 1 m(X),𝒢)PMet X 𝒢. \mathsf{Hom} (\pi^m_1(X), \mathcal{G}) \simeq \mathsf{PMet}^{\mathcal{G}}_X.

As an example, we classify 𝒢\mathcal{G}-torsors over cycle graphs as follows. We note that the notion of a metric group is equivalent to that of a “normed group” as E. Roff stated in her thesis. For a metric group 𝒢\mathcal{G}, we denote the corresponding norm of an element g𝒢g \in \mathcal{G} by |g| 0|g| \in \mathbb{Z}_{\geq 0}.

Proposition  Let C nC_n be an nn-cycle graph. Then we have π 1 m(C n){  with  |1|=1 n: odd, 0 n: even. \pi^m_1(C_n) \cong \begin{cases}\mathbb{Z} &nbsp;\text{ with }&nbsp; |1| = 1 & n : &nbsp;\text{odd}, \\ 0 & n : &nbsp;\text{even}. \end{cases} Hence we have that PMet C n 𝒢{Hom(,𝒢) n: odd, 0 n: even, \mathsf{PMet}_{C_n}^{\mathcal{G}} \simeq \begin{cases}\mathsf{Hom} (\mathbb{Z}, \mathcal{G}) & n : &nbsp;\text{odd}, \\ 0 & n : &nbsp;\text{even}, \end{cases} for any metric group 𝒢\mathcal{G}, which implies that there is only a trivial metric fibration over C 2nC_{2n} and that there is at most one non-trivial metric fibration over C 2n+1C_{2n+1}.

Associated bundle construction and structure groups

Now, similarly to the topological case, we can define an “associated bundle construction” from a torsor and a metric space YY.

First we give another example of a metric group. For a metric space YY, let AutY\mathsf{Aut} Y be the group of self-isometries on YY. We define a distance function on AutY\mathsf{Aut} Y by

d AutY(f,g)=sup yYd Y(fy,gy). d_{\mathsf{Aut} Y}(f, g) = \sup_{y\in Y} d_Y(fy, gy).

Then (AutY,d AutY)(\mathsf{Aut} Y, d_{\mathsf{Aut} Y}) is a metric group if YY is bounded. Even if not, it is an extended metric group that admits \infty as a distance. We suppose that AutY\mathsf{Aut} Y is a metric group in the following, but the discussions can be applied to the extended case.

Now we can construct a metric fibration with fiber YY from a AutY\mathsf{Aut} Y-torsor similarly to the topological case. This construction gives the following.

Theorem  Suppose that YY is a bounded metric space. Then we have a category equivalence PMet X AutYcoreFib X Y, \mathsf{PMet}_X^{\mathsf{Aut} Y} \simeq {\mathsf{core}}\mathsf{Fib}_X^Y, where Fib X Y\mathsf{Fib}_X^Y is the full subcategory of Fib X\mathsf{Fib}_X that consists of metric fibrations with the fiber YY, and we denote the core of a category by core\mathsf{core}.

Čech cohomology of 𝒢\mathcal{G}-torsors

For a 𝒢\mathcal{G}-torsor XX, we define a “11-Čech cohomology” H 1(X,𝒢)\mathsf{H}^1(X, \mathcal{G}) as a groupoid. This is an analogy from the Čech cohomology constructed from the local sections of a principal bundle. We first define the “local section” of a 𝒢\mathcal{G}-torsor.

Definition  Let π:EX\pi : E \longrightarrow X be a 𝒢\mathcal{G}-torsor. For x i,x jXx_i, x_j \in X, we define a local section of π\pi over a pair (x i,x j)(x_i, x_j) as a pair of points (ε i,ε j)E 2(\varepsilon_i, \varepsilon_j) \in E^2 such that πe i=x i,πe j=x j\pi \e_i = x_i, \pi \e_j = x_j and ε j\varepsilon_j is the lift of x jx_j along ε i\varepsilon_i.

We say that ((ε i ij,ε j ij)) (i,j)I 2((\varepsilon^{ij}_i,\varepsilon^{ij}_j))_{(i, j)\in I^2} is a local section of π\pi if each (ε i ij,ε j ij)(\varepsilon^{ij}_i, \varepsilon^{ij}_j) is a local section of π\pi over a pair (x i,x j)(x_i, x_j) and satisfies that ε i ij=ε i ji\varepsilon^{ij}_i = \varepsilon^{ji}_i.

Now we would like to define a cocycle from the differences of any pair of adjacent local sections. Before that, we define the Čech cohomology in general.

Definition  Let XX be a metric space and suppose that points of XX are indexed as X={x i} iIX = \{x_i\}_{i \in I}. For a metric group 𝒢\mathcal{G}, we define the 11-cohomology of XX with coefficients in 𝒢\mathcal{G} as the category H 1(X;𝒢)\mathsf{H}^1(X; \mathcal{G}) by ObH 1(X;𝒢)={(a ijk)𝒢 I 3a ijka kj=a ij,|a ijka jkia kij||Δ(x i,x j,x k)|}, \mathrm{Ob}\mathsf{H}^1(X; \mathcal{G}) = \left\{(a_{ijk}) \in \mathcal{G}^{I^3} \mid a_{ijk}a_{kj\ell } = a_{ij\ell}, |a_{ijk}a_{jki}a_{kij}| \leq |\Delta(x_i, x_j, x_k)|\right\}, and H 1(X;𝒢)((a ijk),(b ijk))={(f ij)𝒢 I 2a ijkf jk=f ijb ijk}, \mathsf{H}^1(X; \mathcal{G})((a_{ijk}), (b_{ijk})) = \left\{(f_{ij}) \in \mathcal{G}^{I^2} \mid a_{ijk}f_{jk} = f_{ij}b_{ijk} \right\}, where we denote the conjugation-invariant norm on 𝒢\mathcal{G} by |||-|. In the above, we used the notation |Δ(x 1,x 2,x 3)||\Delta(x_1, x_2, x_3)| for x 1,x 2,x 3Xx_1, x_2, x_3 \in X defined as |Δ(x 1,x 2,x 3)|:=min{d X(x i,x j)+d X(x j,x k)d X(x i,x k){i,j,k}={1,2,3}}. |\Delta(x_1, x_2, x_3)| := \min \left\{d_X(x_i, x_j) + d_X(x_j, x_k) - d_X(x_i, x_k) \mid \{i, j, k\} = \{1, 2, 3\}\right\}. We call an object of H 1(X;𝒢)\mathsf{H}^1(X; \mathcal{G}) a cocycle. The category H 1(X;𝒢)\mathsf{H}^1(X; \mathcal{G}) is in fact a groupoid.

Proposition  Let π:EX\pi : E \longrightarrow X be a 𝒢\mathcal{G}-torsor. For a local section s=((ε i ij,ε j ij)) (i,j)I 2s =((\varepsilon^{ij}_i,\varepsilon^{ij}_j))_{(i, j)\in I^2} of π\pi, we can construct a cocycle α sπObH 1(X;𝒢)\alpha_s \pi \in \mathrm{Ob} \mathsf{H}^1(X;\mathcal{G}). Further, for any two local sections s,ss, s' of π\pi, the corresponding cocycles α sπ\alpha_s \pi and α sπ\alpha_{s'} \pi are isomorphic.

Conversely we can construct a 𝒢\mathcal{G}-torsor from a cocyle by pasting copies of 𝒢\mathcal{G}’s along the cocycle. Then we have the following from these correspondences.

Theorem   We have a category equivalence H 1(X;𝒢)Tors X 𝒢. \mathsf{H}^1(X; \mathcal{G}) \simeq \mathsf{Tors}^{\mathcal{G}}_X.

Posted at November 23, 2023 10:28 PM UTC

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Re: Classification of Metric Fibrations

There are tons of ideas here! So I’m not sure where to start. But maybe I’ll start near the end, with the definition of the fundamental metric group π 1 m(X,x)\pi_1^m(X, x) of a metric space XX with basepoint xx.

You give as an example that the nn-cycle graph C nC_n has fundamental metric group \mathbb{Z} if nn is odd and the trivial group if nn is even. Can you give some more examples? E.g. what about taking XX to be some other graphs, such as trees? Or what about X=X = \mathbb{R}, or X=S 1X = S^1?

Posted by: Tom Leinster on November 28, 2023 12:14 AM | Permalink | Reply to this

Re: Classification of Metric Fibrations

Tom, thank you for the question!

For trees, R, and geodesic circle S^1, the metric fundamental groups are all trivial! Hence the metric fibrations over them are all trivial. On the other hand, for circle with induced metric from the Euclidean space, it is like a free group with uncountably many generators.

Posted by: Yasuhiko Asao on November 28, 2023 6:28 AM | Permalink | Reply to this

Re: Classification of Metric Fibrations

I see; thanks very much.

I’m still working to understand the general definition, but maybe I can ask one more question.

You’ve mentioned some metric spaces whose ordinary, topological, fundamental group is nontrivial but whose fundamental metric group is trivial.

Do you know of any metric spaces whose ordinary fundamental group is trivial but whose fundamental metric group is nontrivial? Or can this never happen?

Posted by: Tom Leinster on November 28, 2023 5:21 PM | Permalink | Reply to this

Re: Classification of Metric Fibrations

Based only on the examples Yasuhiko gave above, I’ll venture a guess at an example: an arc of a circle with the induced Euclidean metric.

Posted by: Mark Meckes on November 29, 2023 12:40 AM | Permalink | Reply to this

Re: Classification of Metric Fibrations

Yes, it’s correct. An arc is topologically contractible but has non-trivial metric pi1. Also, a sphere with standard Riemannian metric is topologically simply connected but has non-trivial pi1.

I think finding a metric fibration in nature is a bit hard. It is because metric fibration is define to be locally a l^1 product, while wild objects usually have l^2 structure. For example, the standard projection of a plane to a line is not a metric fibration. Once I tried to change the monoidal structure of positive reals from l^1 to l^2 and develop the analogue of metric fibration, but it became too complicated to handle, at least at that time.

Posted by: Yasuhiko Asao on November 29, 2023 5:48 AM | Permalink | Reply to this

Re: Classification of Metric Fibrations

I agree, there are tons of ideas here and I also don’t know where to start, but it looks very interesting. I’m looking forward to hearing you talk about this work next week, Yasuhiko, and I hope I’ll have a good enough handle on it after that to start asking questions!

I suppose the most basic problem is to find more examples of metric fibrations that arise “in the wild”, other than the graph-theoretic examples above (which are already interesting).

Posted by: Mark Meckes on November 28, 2023 3:42 PM | Permalink | Reply to this

Re: Classification of Metric Fibrations

Thanks a lot! And I’m sorry I posted doubly the same thing.

Posted by: Yasuhiko Asao on November 29, 2023 5:52 AM | Permalink | Reply to this

Re: Classification of Metric Fibrations

There is the fundamental metric group π 1 m(X,x)\pi_1^m(X, x) for metric spaces, which are the analogue of the fundamental group for topological spaces. Are there higher homotopy metric groups π n m(X,x)\pi_n^m(X, x) for metric spaces, which are the analogues of the higher homotopy groups π n(X,x)\pi_n(X, x) for topological spaces? And similarly, is there a fundamental metric (infinity) groupoid for metric spaces, which are the analogue of the fundamental (infinity) groupoid for topological spaces?

Posted by: Madeleine Birchfield on December 3, 2023 2:30 PM | Permalink | Reply to this

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