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Yuki Imamura

Publications and source records attributed to Yuki Imamura.

4 recordsLinked to original sources
arXiv2026

Relativization of symmetries on quandles

This paper introduces relative versions of the inner automorphism group and the transvection group associated with surjective quandle homomorphisms.By using the relative inner automorphism group, we define a notion of \emph{connectedness} for surjective homomorphisms. We characterize connected homomorphisms algebraically as quotient maps, and use the relative transvection group to establish a maximal \emph{connected-covering} factorization for arbitrary surjections. Finally, we study surjective homomorphisms for which the relative inner automorphism group acts $2$-transitively on each fiber. Under this assumption, we classify the possible quandle structures of the finite fibers.

math.GT↗
arXiv2024

A formal category theoretic approach to the homotopy theory of dg categories

We introduce a bicategory that refines the localization of the category of dg categories with respect to quasi-equivalences and investigate its properties via formal category theory. Concretely, we first introduce the bicategory of dg categories $\mathsf{DBimod}$, whose Hom categories are given by the derived categories of dg bimodules, and then define the desired bicategory as the sub-bicategory $\mathsf{DBimod}^\text{rqr}$ consisting of right quasi-representable dg bimodules. The first half of the paper is devoted to the study of adjunctions and equivalences in these bicategories. We then show that the embedding $\mathsf{DBimod}^\text{rqr} \hookrightarrow \mathsf{DBimod}$ forms a proarrow equipment in the sense of Richard J. Wood, which provides a framework for formal category theory and enables us to define (weighted) (co)limits in an abstract setting. From this proarrow equipment, we derive the notion of homotopical (co)limits in dg categories, including homotopical shifts and cones, which in turn allows us to give a formal characterization of pretriangulated dg categories. As an application, we provide a conceptual proof of the fact that pretriangulatedness is preserved under the gluing procedure, and we establish reflection results concerning adjoints and colimits.

math.CT↗
arXiv2021

Grothendieck enriched categories

In this paper, we introduce the notion of Grothendieck enriched categories for categories enriched over a sufficiently nice Grothendieck monoidal category $\mathcal{V}$, generalizing the classical notion of Grothendieck categories. Then we establish the Gabriel-Popescu type theorem for Grothendieck enriched categories. We also prove that the property of being Grothendieck enriched categories is preserved under the change of the base monoidal categories by a monoidal right adjoint functor. In particular, if we take as $\mathcal{V}$ the monoidal category of complexes of abelian groups, we obtain the notion of Grothendieck dg categories. As an application of the main results, we see that the dg category of complexes of quasi-coherent sheaves on a quasi-compact and quasi-separated scheme is an example of Grothendieck dg categories.

math.CT↗