In this paper, we consider various categories of hyperbolic Riemann surfaces and show, in various cases, that the conformal or quasiconformal structure of the Riemann surface may be reconstructed, up to possible confusion between holomorphic and anti-holomorphic structures, in a natural way from such a category. The theory exposed in the present paper is motivated partly by a classical result concerning the categorical representation of sober topological spaces, partly by previous work of the author concerning the categorical representation of arithmetic log schemes, and partly by a certain analogy with $p$-adic anabelian geometry --- an analogy which the theory of the present paper serves to render more explicit.
"Conformal and quasiconformal categorical representation of hyperbolic Riemann surfaces." Hiroshima Math. J. 36 (3) 405 - 441, November 2006. https://doi.org/10.32917/hmj/1171377082