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I recently wondered what are the spaces whose morphisms are Lipschitz maps (by which I mean: "locally Lipschitz"). The answer seems pretty clear, and proceeds like the definition of manifolds:

1) If $X$ is a topological space, a Lipschitz chart is a homeomorphism from an open subset of $X$ to a metric space. 2) Two Lipschitz charts are compatible if the corresponding two transition functions are Lipschitz. 3) A Lipschitz atlas on $X$ is a set of compatible Lipschitz charts whose domains cover $X$. 4) A Lipschitz space is a topological space equipped with a maximal Lipschitz atlas.

So my question is: what are these spaces called? (and why not "Lipschitz spaces"?). I'll be grateful for any reference.

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    $\begingroup$ "objects whose morphisms" and "spaces whose morphisms" both sound weird... $\endgroup$
    – YCor
    Commented Dec 27, 2016 at 12:12
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    $\begingroup$ Lipschitz manifolds is quite common. $\endgroup$
    – Pietro Majer
    Commented Dec 27, 2016 at 13:43
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    $\begingroup$ Outside of pathological cases it will generally be possible to find a single metric on $X$ whose restriction to each chart is bi-Lipschitz equivalent to the metric on that chart. So unless you care about the pathological exceptions, you're just talking about metric spaces. "Lipschitz spaces" is not a good term because it is already used to refer to spaces of Lipschitz functions on a metric space, which are functional analytic objects. $\endgroup$
    – Nik Weaver
    Commented Dec 27, 2016 at 16:12
  • $\begingroup$ @Nik You're right, and "non pathological" probably means "paracompact Hausdorff", here, but it seems unnatural to single out a particular metric. The space I'm actually interested in (a sort of "space of shapes") might actually be a Lipschitz Banach manifold, but this looks hard to prove, and I only need a well-defined notion of Lipschitz map into it. $\endgroup$
    – Benoit Jubin
    Commented Dec 27, 2016 at 17:17
  • $\begingroup$ It sounds like what you want is something like "metric space, up to bi-Lipschitz modification"? Maybe uniform structure is what you want, then, although the natural maps are then uniform, not Lipschitz. $\endgroup$
    – Nik Weaver
    Commented Dec 27, 2016 at 18:03

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I randomly found an answer to my question two years later! The spaces I described in the question are called "locally metric spaces" in

J. Luukkainen, J. Väisälä, Elements of Lipschitz topology, Ann. Acad. Sci. Fennicae 3 (1977), 85--122.

They are introduced, together with "local metrics", in their Section 3.4. As guessed in the comments above, any paracompact Hausdorff locally metric space is indeed a metric space "up to Lipschitz equivalence" (their Theorem 3.5). They cite

J. H. C. Whitehead, Manifolds with transverse fields in Euclidean space, Ann. Math. 73 (1961), 154--212.

which introduces "local metrics" in Section 2.

Note: to reply to YCor's comment: indeed, I should have written "objects of the category whose morphisms" instead of "objects whose morphisms".

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    $\begingroup$ Link to paper: acadsci.fi/mathematica/Vol03/vol03pp085-122.pdf $\endgroup$
    – YCor
    Commented Jan 20, 2019 at 18:21
  • $\begingroup$ To be complete: a local metric on a topological space is defined as the data of an open covering $(U_i)$, a metric $d_i$ on each $U_i$, such that the identity map $(U_i\cap U_j,d_i)\to (U_i\cap U_j,d_j)$ is Lipschitz for each $i$. There is a natural equivalence of local metrics (local bilipschitz equivalence). A locally metric space is defined there as a topological space endowed with an equivalence class of local metrics. Of course it's important to emphasize "local" since many people are interested in global or even large-scale aspects of Lipschitz geometry. $\endgroup$
    – YCor
    Commented Jan 20, 2019 at 18:27

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