For the general question about how categorical concepts look when applied to metric spaces, one place to look is Lawvere's paper 'Taking categories seriously', section 6 onwards.
Aside from that, here are a few examples.
Functor categories became function spaces with the uniform or sup metric. That is, if AA and BB are metric spaces construed as enriched categories, then the functor category BABA is the set of distance-decreasing maps AtoBAtoB with the sup metric. (I use "decreasing" in the non-strict sense.)
The (cartesian) product AtimesBAtimesB of two metric spaces --- that is, their product in the category of metric spaces --- has the 'inftyinfty-metric':
d((a,b),(a′,b′))=maxd(a,a′),d(b,b′).d((a,b),(a′,b′))=maxd(a,a′),d(b,b′).
The same goes for infinite products --- remembering that inftyinfty is allowed as a distance. Once you know this, limits in general work in the obvious way.
The coproduct A+BA+B of two metric spaces AA and BB is their disjoint union, with d(a,b)=d(b,a)=inftyd(a,b)=d(b,a)=infty for all ainA,binBainA,binB. Again, it's crucial here to allow inftyinfty as a distance. Otherwise, your category of metric spaces will lack lots of limits and colimits. The coequalizer of two maps f,g:AtoBf,g:AtoB is BB quotiented out by the usual equivalence relation simsim (as in the category of sets), and metrized by
d([b],[b′])=infd(y,y′):ysimb,y′simb′d([b],[b′])=infd(y,y′):ysimb,y′simb′
where [b][b] denotes the equivalence class of bb. General colimits work similarly.
I mentioned the cartesian product, but there's another kind of product. Generally, if mathbfVmathbfV is a monoidal category then any two mathbfVmathbfV-enriched categories, AA and BB, have a tensor product AotimesBAotimesB. Its set of objects is the product of the sets of objects of AA and BB. Its hom-objects are given by
(AotimesB)((a,b),(a′,b′))=A(a,a′)otimesB(b,b′).(AotimesB)((a,b),(a′,b′))=A(a,a′)otimesB(b,b′).
This gives us a tensor product of metric spaces. Given metric spaces AA and BB, the point-set of AotimesBAotimesB is the product of the point-sets of AA and BB. The distance is given by
d((a,b),(a′,b′))=d(a,a′)+d(b,b′).d((a,b),(a′,b′))=d(a,a′)+d(b,b′).
In other words, it's the '11-metric', also known as the taxicab metric, Manhattan metric, etc.
So, Andrew, when you ask 'What is a monoidal metric space?', you have to say which product you want to be monoidal with respect to. That is, are you asking about (weak) monoids in (mathbfMet,times)(mathbfMet,times) or in (mathbfMet,otimes)(mathbfMet,otimes)?
From the tone of your question, I would guess: both. So here goes.
The answer for cartesian product timestimes doesn't seem so interesting. Assuming that your metric spaces satisfy the classical skeletality axiom (d(a,b)=0Rightarrowa=bd(a,b)=0Rightarrowa=b), a monoidal category for timestimes is a metric space AA equipped with a monoid structure on its set of points such that
d(acdotb,a′cdotb′)leqmaxd(a,a′),d(b,b′).d(acdotb,a′cdotb′)leqmaxd(a,a′),d(b,b′).
I can't think of anything more to say about that.
The answer for tensor product otimesotimes seems more interesting. A monoid in (mathbfMet,otimes)(mathbfMet,otimes) is a metric space AA equipped with a monoid structure on its set of points such that for all aa, the maps acdot−acdot− and −cdota−cdota are distance-decreasing. For example, if it's a group then this says that left or right translation is always an isometry. This often happens: consider the underlying additive group of a normed vector space, for instance.
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