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Local Categories and Metric Sites: The concept of a metric topology formalizes the passage from spectrums to schemes (i.e. the completion a strict metric site), as well as the theory of fibre products of schemes. After finishing  On the Geometry of Metric Sites, which was later published in J. Algebra in 1995, I found a copy of S. Eilenberg's lecture notes Foundation of Fibre Bundles (University of Chicago), based on the joint work with H. Cartan, in which  the notion of a local category is given. Basically a local category is a category together with a functor to the category of topological spaces such that the pullback sieve of any open sieve is represented by a mono whose image is an open embedding, and the map between objects can be defined locally. Thus a local category is essentially a strict effective metric site. Their theorem 5.1 of  the "2^nd enlargement" of  a local category is a special case of the completion of a strict metric site. (Top of Page)

Unitary Categories: In the summer of 1996, I wrote two papers on categorical geometry, both based on Diers's book on Zariski categories. One is Unitary Categories, the other one eventually became the first two chapters of Categorical Geometry. The first one is from algebraic point view, and the main notion is an ideal (i.e. congruence). The second one is from geometric point view, and the main notion is a regular mono. Since there is a Galois correspondence between ideals and regular monos (at least when the category is an analytic geometry such that any strong mono is regular), the invariants (such as spectrums in Introduction, or spectrums in Prime Ideals) derived from these two approaches are essentially the same. This is one of the main theorem of categorical geometry.