Enriched Categories: Metric Spaces, Monoidal Endofunctors and Lipschitz-Continuous Maps.

2011-12-20T14:09:27Z

In the introduction to the reprint of "Metric spaces, generalized logic and closed categories" Lawvere talks about the following situation:

Let $\mathbb R_+$ denote $\mathbb R_{\geq 0}^\infty$. Every lax monoidal endofunctor $\lambda:\mathbb R_+\to\mathbb R_+$ gives rise to an endofunctor $\lambda.:\mathrm{Met}\to\mathrm{Met}$ on the category of generalized metric spaces, again leading to the notion of $\lambda$-Lipschitz-continous maps $$ Lip^\lambda(X,Y):=Lip^1(X,\lambda.Y).$$ These are exactly the maps $f:X\to Y$ satisfying $d(x,x')\geq \lambda(d(fx,fx'))$. Lawvere goes on and "suggest a whole family of monoidal structures [on the category of metric spaces] interpolating between" the sum- (aka tensor product) and the max- (aka cartesian product) metric on the product of the underlying sets. He finally relates this to the $$\frac{1}{p}+\frac{1}{q}=1$$ business occuring in analysis.

Now for the question: Has this been worked out somewhere (replacing $\mathbb R_+$ by an arbitrary moinoidal category $\mathcal V$)?

Remark: Concerning the monoidal structures my first idea was to define various adjoints to the hom-like functors $$Lip^\lambda(X,-)$$ but as we don't have $$Lip^\lambda\times Lip^\lambda\to Lip^\lambda$$ but rather $$Lip^\lambda\times Lip^\mu\to Lip^{\mu\circ\lambda}$$ i suspect we should define $X\otimes_\lambda^\mu Y$ by an expression like $$Lip^\lambda(X,Lip^\mu(Y,Z))=:Lip^\lambda(X\otimes_\lambda^\mu Y,Z)$$ (of maybe $Lip^1$ on the right hand side). So instead of various monoidal structures we'd get various tensor products, compatible in some way...

Enriched Categories: Metric Spaces, Monoidal Endofunctors and Lipschitz-Continuous Maps. - MathOverflow

Enriched Categories: Metric Spaces, Monoidal Endofunctors and Lipschitz-Continuous Maps.