An Elementary Account of Flat 2-Functors
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A set-valued functor is “flat” if its tensor product extension is finite-limit preserving. Such a functor is flat if, and only if, its category of elements is filtered. Analogously, a category-valued 2-functor on a 2-category is defined to be flat in terms of a finite-limit preserving property. The characterization in the work of M. E. Descotte, E. J. Dubuc, and M. Szyld is that a 2-functor is flat if, and only if, its 2-category of elements is appropriately 2-filtered. The goal of the present work is to prove a generalization in the internal 2-category theory of a suitable 1-category. This follows the pattern of R. Diaconescu’s generalized account of the theory 1-dimensional flatness in the internal category theory of a 1-topos. The 1-topos is here replaced by the 2-category of internal categories of an exact 1-category. This work follows a novel approach. The first step is in computing, for a category-valued pseudo-functor, a tensor product extension. This is done as a category of fractions. Supposing this extension is finite-limit preserving, 2-filteredness conditions are obtained related to those of Descotte, Dubuc and Szyld. The converse result, namely, that our 2-filteredness conditions imply finite-limit preservation, is approached using the right calculus of fractions. That is, under the assumption of 2-filteredness, the tensor product is formed through a right calculus of fractions. This gives a tractable characterization of the morphisms of the tensor product, from which follows an “elementary” proof that filteredness implies limit-preservation. For the internal generalization, the right calculus of fractions is described in internal cat- egory theory. The internal 2-filteredness conditions imply that an internal tensor-product construction is formed through the internal right calculus of fractions. Finally, it is seen that internal 2-filteredness implies that the internal tensor product is finite-limit preserving. Partly this is achieved by showing that the internal tensor product reduces to Diaconescu’s internal colimit construction. For this reason, exactness of the internal tensor product partly reduces to known cases. The remaining case is that of certain cotensors, which are shown to be preserved using an elementary argument.