Compact convolution

Colimits in enriched $\infty$-categories and Day convolution

For a monoidal $\infty$-category $\mathcal{M}$ with colimits, we study colimits of $\mathcal{M}$-functors $\mathcal{A}\to\mathcal{B}$ where $\mathcal{B}$ is left-tensored over $\mathcal{M}$ and $\mathcal{A}$ is an $\mathcal{M}$-enriched category. We prove that the enriched Yoneda embedding $Y:\mathcal{A}\to P_{\mathcal{M}}(\mathcal{A})$ yields a universal $\mathcal{M}$-functor and, in the case when $\mathcal{A}$ has a certain monoidal structure, the category of enriched presheaves $P_{\mathcal{M}}(\mathcal{A})$ inherits the same monoidal structure.

Biclosed bicategories: localisation of convolution

We give a summary (without proofs) of the main results in the author's thesis entitled ``Construction of biclosed categories'' (University of New South Wales, Australia, 1970). This summary is reprinted directly from Report 81-0030 of the School of Mathematics and Physics, Macquarie University, April 1981. In particular, it gives sufficient conditions for existence of an extension of a (pro)monoidal category structure along a given dense functor to a cocomplete category. The two basic procedures used in the proof turn out to be special cases of the final result, the two respective dense functo…

Characterisations of $Σ$-pure-injectivity in triangulated categories and applications to endoperfect objects

We provide various ways to characterise $Σ$-pure-injective objects in a compactly generated triangulated category. These characterisations mimic analogous well-known results from the model theory of modules. The proof involves two approaches. In the first approach we adapt arguments from the module-theoretic setting. Here the one-sorted language of modules over a fixed ring is replaced with a canonical multi-sorted language, whose sorts are given by compact objects. Throughout we use a variation of the Yoneda embedding, called the resticted Yoneda functor, which associates a multi-sorted struc…

三角范畴之间的可定义函子

Definable functors between triangulated categories

该文系统发展紧生成三角范畴间的可定义函子理论，刻画凝聚函子并给出可定义函子的等价条件，还将结果应用于Ziegler谱的函子性研究。

The convolution algebra of an absolutely locally compact topos

We introduce a class of toposes called "absolutely locally compact" toposes and of "admissible" sheaf of rings over such toposes. To any such ringed topos $(\mathcal{T},A)$ we attach an involutive convolution algebra $\mathcal{C}_c(\mathcal{T},A)$ which is well defined up to Morita equivalence and characterized by the fact that the category of non-degenerate modules over $\mathcal{C}_c(\mathcal{T},A)$ is equivalent to the category of sheaf of $A$-module over $\mathcal{T}$. In the case where $A$ is the sheaf of real or complex Dedekind numbers, we construct several norms on this involutive alge…

扭曲双函子与宽松离散opfibration

Twisted double functors and loosely discrete opfibrations

本文针对双范畴理论及应用中所需的宽松双余预层概念，开发并比较其模型。引入扭曲双函子定义扭曲余预层，展示其例子，证明扭曲余预层与分裂宽松离散opfibration等价，还通过拼贴构造比较扭曲双模与特定双范畴。