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.



Objects whose morphisms are Lipschitz maps
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".


