We discuss a new type of weak factorization system. Although these systems provide (up to isomorphism) uniquely determined decompositions of morphisms, in general they do not constitute orthogonal factorizations and are not even functorial. Nevertheless, they arise naturally, as injective hulls or projective covers in comma categories. Surprisingly, often injective hulls and projective covers c...

We describe an isomorphism of categories conjectured by Kontsevich. If M and M̃ are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya’s category of Lagrangian submanifolds on M̃. We prove this equivalence when M is an elliptic curve and M̃ is its dual curve, exhibiting the dictionary in detail.

Categories of partial functions have become increasingly important principally because of their applications in theoretical computer science. In this note we prove that the category of partial bijections between sets as an inverse-Baer*-category with closed projections and in which the idempotent split is an exact category. Finally the Noether isomorphism theorems are given for this exact categ...

We show that Hofmann’s and Curien’s interpretations of Martin-Löf’s type theory, which were both designed to cure a mismatch between syntax and semantics in Seely’s original interpretation in locally cartesian closed categories, are related via a natural isomorphism. As an outcome, we obtain a new proof of the coherence theorem needed to show the soundness after all of Seely’s interpretation.

Many formulations of proof nets and sequent calculi for Classical Linear Logic (CLL) [7, 8] take it for granted that a type A is identical to its double negation A⊥⊥. On the other hand, since Seely [13], it has been assumed that ∗-autonomous categories [1, 2] are the appropriate semantic models of (the multiplicative fragment of) CLL. However, in general, in a ∗-autonomous category an object A ...

We have been developing a segmentation method for touching pairs of numerals. In this study, we extend the method to the segmentation of numeral strings that contain the touching patterns of multiple numerals. The mechanisms of the separation of overlapping strokes and the removal of ligature are introduced into our “segmentation graph.” We also attempt to combine local features and structural ...

In this article, we develop a theory of Grothendieck’s six operations for derived categories in étale cohomology of Artin stacks. We prove several desired properties of the operations, including the base change theorem in derived categories. This extends all previous theories on this subject, including the recent one developed by Laszlo and Olsson, in which the operations are subject to more as...

We prove the universal property of the infinitary exact completion of a category with weak small limits. As an application, we slightly weaken the conditions characterizing essential localizations of varieties (in particular, of module categories) and of presheaf categories. Introduction An essential localization is a reflective subcategory such that the reflector has a left adjoint. In [12], R...