# Category of metric spaces versus category of non-empty spaces

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, many 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.

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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 4 at 14:39
@egreg could you please be clearer: do you prefer $\mathbf{Met}$ or $\mathbf{Met}_{\not=\emptyset}$ ? –  magma Jul 5 at 11:01
@magma Less restrictive = allow the empty space. –  egreg Jul 5 at 17:08