Postprint version. Published in International Journal of Theoretical Physics, Volume 49, Issue 8, January 1, 2010, pages 1952-1960.
The definitive version is available at https://doi.org/10.1007/s10773-010-0380-8.
We will give descriptions of u-singularities as we introduce the notion of t-topos theoretic entropies. The unifying methodology for a u-singularity is the universal mapping property of an inverse or direct limit. The qualitative, conceptual, and structural analyses of u-singularities are given in terms of inverse and direct limits of micro decompositions of a presheaf and coverings of an object in t-site in the theory of temporal topos.