Does there exist a characterization of Goedel's constructible universe $L$ in purely category-theoretic terms, or is constructibility an 'artifact' of material set theory? If, in fact, constructibility is an 'artifact' of material set theory, is there a category-theoretic 'analogue' for $L$?
Consider the following statement from Lawvere's paper, "Cohesive Toposes and Cantor's 'lauter Einsen' (Philosophia Mathematicae (3) Vol. 2 (1994), pp. 5-15):
It might seem that G$\ddot o$del's $L$ would rest, if anything does, on von Neumann's a priori $\epsilon$-chains, but that was refuted twenty years ago by William Mitchell who showed how to construct it from a suitable given category, structured only by composition of maps; however, to my knowledge this has not been pursued since.
Though it might not be 'kosher' to answer a question with a question, why does Mitchell's construction not provide the necessary category-theoretic characterization of $L$ I seek (Mitchell's construction is contained in his papers, "Boolean Topoi and the Theory of Sets" (Journal of Pure and Applied Algebra 2(1972), 261-274) and "Categories of Boolean Topoi" (Journal of Pure and Applied Algebra 3 (1973) 193-201))?