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Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have $$d(x,y)\ge d'(fx,fy)\,.$$ Let $\mathbf{Met}_{\not=\emptyset}$ be the full subcategory of $\mathbf{Met}$ with non-empty spaces as objects.

The category $\mathbf{Met}$ is, what Wikipedia defines as the category of metric spaces. However, some analysis books require metric spaces to be non-empty.

My question is, which of the above two categories is more well-behaved, and which definition should therefore be preferred from a categorical point of view.

For example, $\mathbf{Met}_{\not=\emptyset}$ has no initial object, whereas $\mathbf{Met}$ does, namely the empty space which uniquely embeds into every other metric space. Are there other differences concerning existence of limits and colimits? Are arrows in $\mathbf{Met}_{\not=\emptyset}$ (split/effective/descent/regular/extremal) monomorphisms and epimorphisms iff their images under the embedding functor to $\mathbf{Met}$ are, or do the characterizations change?

Note that this question is not a duplicate of SE/45145, since I am not merely interested in topological but rather in categorical aspects.

Martin Brandenburg
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    I'd go with the less restrictive definition. There is no morphism from a nonempty space to the empty space, so, for instance, the characterization of a split morphism cannot change. – egreg Jul 04 '14 at 14:39
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    @egreg could you please be clearer: do you prefer $\mathbf{Met}$ or $\mathbf{Met}_{\not=\emptyset}$ ? – magma Jul 05 '14 at 11:01
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    @magma Less restrictive = allow the empty space. – egreg Jul 05 '14 at 17:08
  • Why is the metric space with a single element not the initial object for $\mathbf{Met}_{\not=\phi}$? What am I missing? – N Unnikrishnan Jan 08 '17 at 10:22
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    @NUnnikrishnan Because there are lots of maps ${\ast}\longrightarrow X$ for general $X$. –  Jan 13 '17 at 18:23
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    Which analysis books prohibit the empty space? – HeinrichD Apr 05 '17 at 09:13
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    "My question is, which of the above two categories is more well-behaved" <--- You have already answered this question yourself. – Martin Brandenburg Jan 24 '25 at 06:28
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    If you have reasons to exclude the empty space you might consider the category of pointed metric spaces (with morphisms preserving base points). Then you even have objects which are initial and terminal. – Jochen Jan 24 '25 at 10:09

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