Tobias Fritz
The theory of categories of fractions as originally developed by Gabriel and Zisman is reviewed in a pedagogical manner giving detailed proofs of all statements. A weakening of the category of fractions axioms used by Higson is discussed and shown to be equivalent to the original axioms.
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Sieg and Wegner showed that the stable exact sequences define a maximal exact structure (in the sense of Quillen) in any pre-abelian category. We generalize this result for weakly idempotent complete additive categories.
We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over Cartesian categories, so that every Cartesian category has an associated cofree differential category. We also work out the corresponding results when the categories involved have restriction structure, and show that these categories are closed under splitting restriction idempotents.
Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full subcategories of strong and weakly topologized objects and show that each is equivalent to the chu category of the original category with respect to the dualizing object.
Products in double categories, revisited
本文从无偏视角重新审视双范畴积，提出自由积完成的双范畴类比，扩展了双范畴的表达能力，同时保持单一普遍性质，并通过双重性发展积与余积理论及有限积双理论应用。
Cartesian Linearly Distributive Categories: Revisited
本文重新审视笛卡尔线性分配范畴，证明其强结构性质，并研究了正则分配范畴和半加性范畴等关键例子，同时指出某些之前认为是CLDCs的类别实际上并非如此。
Consider an exact couple in a semiabelian category in the sense of Palamodov, i.e., in an additive category in which every morphism has a kernel as well as a cokernel and the induced morphism between coimage and image is always monic and epic. Assume that the morphisms in the couple are strict, i.e., they induce even isomorphisms between their corresponding coimages and images. We show that the classical construction of Eckmann and Hilton in this case produces two derived couples which are connected by a natural bimorphism. The two couples correspond to the a priori distinct cohomology objects…
Homotopy limits of model categories, revisited
本文回顾模型范畴左Quillen函子图表同伦极限的定义及应用，探讨改用右Quillen函子的可行性，指出无法同时使用左右Quillen函子的原因。
After recalling basic definitions and constructions for a finite group $G$ action on a $k$-linear category we give a concise proof of the following theorem of Elagin: if $\mathcal{C} = \langle \mathcal{A}, \mathcal{B} \rangle$ is a semiorthogonal decomposition of a triangulated category which is preserved by the action of $G$, and $\mathcal{C}^G$ is triangulated, then there is a semiorthogonal decomposition $\mathcal{C}^G = \langle \mathcal{A}^G, \mathcal{B}^G \rangle$. We also prove that any $G$-action on $\mathcal{C}$ is weakly equivalent to a strict $G$-action which is the analog of the Coh…
Differential categories were introduced to provide a minimal categorical doctrine for differential linear logic. Here we revisit the formalism and, in particular, examine the two different approaches to defining differentiation which were introduced. The basic approach used a deriving transformation, while a more refined approach, in the presence of a bialgebra modality, used a codereliction. The latter approach is particularly relevant to linear logic settings, where the coalgebra modality is monoidal and the Seely isomorphisms give rise to a bialgebra modality. Here, we prove that these appa…
Bicartesian Coherence Revisited
综述了关于有限积范畴和有限余积范畴的凝聚性的研究结果。对于这些结果，这是以前发表的作者在几个地方，一些配方和证明在这里更正，并更新事项。本文研究的范畴形式化了经典和直觉合取-析取逻辑中的证明的等价性，而没有合取在析取上的分布。
We prove that the folk model structure on strict $\infty$-categories transfers to the category of strict $\infty$-groupoids (and more generally to the category of strict $(\infty, n)$-categories), and that the resulting model structure on strict $\infty$-groupoids coincides with the one defined by Brown and Golasinski via crossed complexes.
This paper adresses two issues in dealing with bicategories of fractions. The first is to introduce a set of conditions on a class of arrows in a bicategory which is weaker than the one given in Pronk, Etendues and stacks as bicategories of fractions, but still allows a bicalculus of fractions. These conditions allow us to invert a smaller collection of arrows so that in some cases we may obtain a bicategory of fractions with small hom-categories. We adapt the construction of the bicategory of fractions to work with the weaker conditions. The second issue is the difficulty in dealing with 2-ce…
Hopfological algebra, revisited
本文提出一种∞-范畴化方法处理Khovanov–Qi的Hopfological代数，通过将先前构造重铸为幺半∞-范畴中的模∞-范畴，精炼了理论的基础方面，并推广到任意刚性紧生成对称幺半稳定∞-范畴上。
Exactness of limits and colimits in abelian categories revisited
本文重新审视了在交换范畴中极限和余极限的精确性，证明了相关函子的精确性与Ext-伴随性之间的关系，并提供了新的证明方法。
Pronk's theorem on bicategories of fractions is applied, in almost all cases in the literature, to 2-categories of geometrically presentable stacks on a 1-site. We give an proof that subsumes all previous such results and which is purely 2-categorical in nature, ignoring the nature of the objects involved. The proof holds for 2-categories that are not (2,1)-categories, and we give conditions for local essential smallness.
We study the interaction between the notions of filteredness, fractions and fibrations in the theory of bicategories, generalizing classical results for categories. We give an explicit formula for filtered pseudo-colimits of categories indexed by a bicategory, and we use it to compute the hom-categories of a bicategory of fractions. As a consequence, we show that the canonical pseudo-functor into a bicategory of fractions is exact.
We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor Hom(T, -), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equivalent to the category of finite dimensional modules over the endomorphism algebra of T in C.
For a composition-closed and pullback-stable class S of morphisms in a category C containing all isomorphisms, we form the category Span(C,S) of S-spans (s,f) in C with first "leg" s lying in S, and give an alternative construction of its quotient category C[S^{-1}] of S-fractions. Instead of trying to turn S-morphisms "directly" into isomorphisms, we turn them separately into retractions and into sections in a universal manner, thus obtaining the quotient categories Retr( C,S) and Sect(C,S). The fraction category C[S^{-1}] is their largest joint quotient category. Without confining S to be a …
We study the *homotopy theory* of $\infty$-categories enriched in the $\infty$-category $sS$ of simplicial spaces. That is, we consider $sS$-enriched $\infty$-categories as presentations of ordinary $\infty$-categories by means of a "local" geometric realization functor $Cat_{sS} \to Cat_\infty$, and we prove that their homotopy theory presents the $\infty$-category of $\infty$-categories, i.e. that this functor induces an equivalence $Cat_{sS} [[ W_{DK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$ from a localization of the $\infty$-category of $sS$-enriched $\infty$-categories. Following Dwyer--Ka…
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