Limits and Colimits in the Category of Pastures

A construction of certain weak colimits and an exactness property of the 2-category of categories

Given a 2-category $\mathcal{A}$, a $2$-functor $\mathcal{A} \overset {F} {\longrightarrow} \mathcal{C}at$ and a distinguished 1-subcategory $Σ\subset \mathcal{A}$ containing all the objects, a $σ$-cone for $F$ (with respect to $Σ$) is a lax cone such that the structural $2$-cells corresponding to the arrows of $Σ$ are invertible. The conical $σ$-limit is the universal (up to isomorphism) $σ$-cone. The notion of $σ$-limit generalises the well known notions of pseudo and lax limit. We consider the fundamental notion of $σ$-filtered} pair $(\mathcal{A}, \, Σ)$ which generalises the notion of 2-f…

内部高阶范畴论中的余极限与余完备化

Colimits and cocompletions in internal higher category theory

本文发展了以 $\infty$-topos 为内部的范畴理论，讨论了内部范畴的伴随、极限、余极限与 Kan 扩张，证明了内部预层范畴的泛性质，并构造了由任意图形形状类索引的自由余完备化。

限制类别III：共极限、部分极限和扩展性

Restriction categories III: colimits, partial limits, and extensivity

限制范畴是部分映射范畴的一个抽象公式，定义为某些特定的幂等元，称为限制幂等元。部分映射的所有范畴都是限制范畴;相反，限制范畴是部分映射的范畴当且仅当限制幂等元分裂。限制范畴促进了关于部分映射的推理，因为它们具有纯代数形式。本文研究了限制范畴中的余极限和极限。由于限制范畴的概念不是自对偶的，我们不应该期望限制范畴中的余极限和极限以相同的方式表现。限制上下文中的余限概念非常简单，但极限更微妙。适当的极限概念是一种宽松的极限，满足某些额外的性质。特别令人感兴趣的是联积本身的行为和关于部分乘积的行为。我们探讨了各种条件下的余产品是“广泛的”在这个意义上说，总范畴（相关的部分映射范畴）成为广泛的范畴。当部分极限存在时，它们就变成了总范畴中的普通极限。因此，当余积是广延的时，我们得到一个广延范畴作为全范畴。特别是，这提供了一个描述广泛完成的分配范畴。

∞-拓扑斯的∞-范畴中的余极限与平展态射

Colimits in the $\infty$-category of $\infty$-topoi and étale morphisms

该研究提出Lurie关于∞-拓扑斯∞-范畴中平展态射宽子范畴对小余极限封闭结论的新证明，基于平展态射与泛性质族的新刻画且不依赖更大宇宙，还给出泛性质完备化的初等构造。

定向范畴理论中的余极限

Colimits in Oriented Category Theory

本文针对高阶范畴理论中lax余极限的几何不兼容性缺陷，提出定向余极限理论，引入兼容Gray张量积的Grothendieck构造，证明其等价性并将其应用于高阶主丛分类、伴随函子表示及Quillen定理推广。

元素双范畴的 Thomason 余极限定理

Thomason's colimit theorem for the double category of elements

本文将 Thomason 余极限定理推广到 Grandis–Paré 元素双范畴，证明其分类空间与 2-函子同伦余极限的分类空间弱同伦等价。

Limits and colimits of quantaloid-enriched categories and their distributors

It is shown that, for a small quantaloid Q, the category of small Q-categories and Q-functors is total and cototal, and so is the category of Q-distributors and Q-Chu transforms.

作为2-胞腔的内自同构

Inner automorphisms as 2-cells

本研究利用抽象内自同构将普通范畴提升为2-范畴，分析了所得2-范畴中各类二维极限、余极限的存在条件，并探讨了普通函子延拓为2-函子的情形。

Left fibrations and homotopy colimits II

For a small simplicial category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the homotopy-coherent nerve of A provides a left Quillen equivalence between the projective model structure on the former category and the covariant model structure on the latter. We compare this Quillen equivalence to the straightening-unstraightening equivalence previously established by Lurie, where the left adjoint goes in the opposite direction. The existence of left Quillen functors in both directions considerably simplifies t…

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.

在 Cat 中形式化余极限

Formalizing colimits in Cat

本文在 Lean 中形式化了范畴的 1-范畴的基本构造，构造神经嵌入的左伴随并证明该伴随是反射的，从而首次形式化证明 Cat 具有余极限且范畴的范畴是余完备的。

Colimits of monoids

If $\mathcal{C}$ is a cocomplete monoidal category in which tensoring from both sides preserves coequalizers, then the category of monoids over $\mathcal{C}$ is cocomplete. The same holds if $\mathcal{C}$ has regular factorizations and tensoring only preserves regular epimorphisms. As an application a lifting theorem for an adjunction with a monoidal right adjoint to an adjunction between the respective categories of monoids is proved.

A Construction of Colimits in Monoidal Closed Categories

We prove that a category which is symmetric (relaxed) monoidal closed, (small) complete, well-powered and has a small cogenerating family, is cocomplete.

Diagrams, Fibrations, and the Decomposition of Colimits

The contributions of this paper are twofold. Within the framework of Grothendieck's fibrational category theory, we present a web of fundamental 2-adjunctions surrounding the formation of the category of all small diagrams in a given category and the formation of the Grothendieck category of a functor into the category of small categories. We demonstrate the utility of these adjunctions, in part by deriving three formulae for (co-)limits: a `twisted' generalization of the well-known Fubini formula, as first established by Chachólski and Scherer; a new `general colimit decomposition formula'; a…

A colimit decomposition for homotopy algebras in Cat

Badzioch showed that in the category of simplicial sets each homotopy algebra of a Lawvere theory is weakly equivalent to a strict algebra. In seeking to extend this result to other contexts Rosicky observed a key point to be that each homotopy colimit in simplicial sets admits a decomposition into a homotopy sifted colimit of finite coproducts, and asked the author whether a similar decomposition holds in the 2-category of categories Cat. Our purpose in the present paper is to show that this is the case.

Marked colimits and higher cofinality

Given a marked $\infty$-category $\mathcal{D}^{\dagger}$ (i.e. an $\infty$-category equipped with a specified collection of morphisms) and a functor $F: \mathcal{D} \to \mathbb{B}$ with values in an $\infty$-bicategory, we define $\operatorname{colim}^{\dagger} F$, the marked colimit of $F$. We provide a definition of weighted colimits in $\infty$-bicategories when the indexing diagram is an $\infty$-category and show that they can be computed in terms of marked colimits. In the maximally marked case $\mathcal{D}^{\sharp}$, our construction retrieves the $\infty$-categorical colimit of $F$ in …

Homotopy weighted colimits

Let V be a cofibrantly generated closed symmetric monoidal model category and M a model V-category. We say that a weighted colimit W*D of a diagram D weighted by W is a homotopy weighted colimit if the diagram D is pointwise cofibrant and the weight W is cofibrant in the projective model structure on [C^op,V]. We then proceed to describe such homotopy weighted colimits through homotopy tensors and ordinary (conical) homotopy colimits. This is a homotopy version of the well known isomorphism W*D=\int^C(W\tensor D). After proving this homotopy decomposition in general we study in some detail a f…

A characterisation of algebraic exactness

An algebraically exact category in one that admits all of the limits and colimits which every variety of algebras possesses and every forgetful functor between varieties preserves, and which verifies the same interactions between these limits and colimits as hold in any variety. Such categories were studied by Adámek, Lawvere and Rosický: they characterised them as the categories with small limits and sifted colimits for which the functor taking sifted colimits is continuous. They conjectured that a complete and sifted-cocomplete category should be algebraically exact just when it is Barr-exac…

Adjoining colimits

This paper develops a theory of colimit sketches "with constructions" in higher category theory, formalising the input to the ubiquitous procedure of adjoining specified "constructible" colimits to a category such that specified "relation" colimits are enforced (or preserved). From a more technical standpoint, sketches are a way to describe dense functors using techniques from the homotopy theory of diagrams. We establish basic properties of diagrams in an infinity-category C as a model for presheaves on C and Bousfield localisations thereof, discuss extensions of functors and adjunctions, and…

重新审视 $\mathbf{Cat}$ 中的极限和同伦范畴

Revisiting colimits in $\mathbf{Cat}$ and homotopy category

本文通过明确的方法证明了 $\mathbf{Cat}$ 中 (co) 限的存在，通过同伦范畴 functor 与特定加权极限的关系，以及利用 simplicial sets 和 nerve functor 的性质，证明了 $\mathbf{Cat}$ 的 (co) 完备性。

Lex colimits

Many kinds of categorical structure require the existence of finite limits, of colimits of some specified type, and of "exactness" conditions between the finite limits and the specified colimits. Some examples are the notions of regular, or Barr-exact, or lextensive, or coherent, or adhesive category. We introduce a general notion of exactness, of which each of the structures listed above, and others besides, are particular instances. The notion can be understood as a form of cocompleteness "in the lex world" -- more precisely, in the 2-category of finitely complete categories and finite-limit…

Realizable homotopy colimits

In this paper we prove that for any model category, the Bousfield-Kan construction of the homotopy colimit is the absolute left derived functor of the colimit. This is achieved by showing that the Bousfield-Kan homotopy colimit is moreover a realizable homotopy colimit, defined by means of a suitable 2-category of relative categories. In addition, in the case of exact coproducts, we characterize the realizable homotopy colimits that satisfy a cofinality property as those given by a formula following the pattern of Bousfield-Kan construction: they are the composition of a "geometric realization…

二维切片中的余极限

Colimits in 2-dimensional slices

本研究在增强范畴论框架下将一维切片余极限理论推广到二维，定义lax切片与相关2-函子，证明其2-余极限的保持、反射与提升性质，并将基变换2-函子的余极限保持结果推广到带稠密生成子的有限完全2-范畴。

Note on filtered colimits of Hilbert spaces

The category of Hilbert spaces and contractions has filtered colimits, and tensoring preserves them. We also discuss (problems with) bounded maps.

参数化高阶范畴论

Parametrized higher category theory

本文为参数化∞-范畴建立范畴论基础，提出索引同伦极限与余极限理论，并证明G-空间范畴由可缩G-空间自由生成，证实Hill猜想。

Homotopy colimits in stable representation theory

We study the problem of existence and uniqueness of homotopy colimits in stable representation theory, where one typically does not have model category structures to guarantee that these homotopy colimits exist or have good properties. We get both negative results (homotopy cofibers fail to exist if there exist any objects of positive finite projective dimension!) and positive results (reasonable conditions under which homotopy colimits exist and are unique, even when model category structures fail to exist). Along the way, we obtain relative-homological-algebraic generalizations of classical …

朴素离散同伦理论中余极限的不存在性

Nonexistence of colimits in naive discrete homotopy theory

本文证明简单图范畴在A-同伦等价类上局部化所得拟范畴不存在余极限，从而否定了A-同伦等价构成模型结构一部分的猜想。

通过加权余极限的余偏序

Cofinality via Weighted Colimits

本文通过加权余极限对偶现象，推广了Quillen定理A，并给出了在稳定有理∞-范畴中自由∞-代数的简化公式。

Homotopy colimits of model categories

Building on a previous definition of homotopy limit of model categories, we give a definition of homotopy colimit of model categories. Using the complete Segal space model for homotopy theories, we verify that this definition corresponds to the model-category-theoretic definition in that setting.

Stone duality for topological theories

In the context of categorical topology, more precisely that of T-categories [Hofmann, 2007], we define the notion of T-colimit as a particular colimit in a V-category. A complete and cocomplete V-category in which limits distribute over T-colimits, is to be thought of as the generalisation of a (co-)frame to this categorical level. We explain some ideas on a T-categorical version of "Stone duality", and show that Cauchy completeness of a T-category is precisely its sobriety.

Left fibrations and homotopy colimits

For a small category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the nerve of A establishes a left Quillen equivalence between the projective (or Reedy) model structure on the former category and the covariant model structure on the latter. We compare this equivalence to a Quillen equivalence in the opposite direction previously established by Lurie. From our results we deduce that a categorical equivalence of simplicial sets induces a Quillen equivalence on the corresponding over-categories, equipped with …

A Probability Monad as the Colimit of Spaces of Finite Samples

We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein distance. This monad is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1-bounded complete metric spaces. We prove that this Kantorovich monad arises from a colimit construction on finite power-like constructions, which formalizes the intuition that probability measures are limit…

小函子的极限

Limits of small functors

对于一个在适当的幺半范畴V上富集的小范畴K，K在余极限下的自由完备化是预层范畴[K*，V].如果K是大的，则它在上极限下的自由完备化是K上小预层的V-范畴PK，其中一个预层是小的，如果它是一个具有小Domain的预层的左Kan扩张。我们研究了PK上极限的存在性和么半群闭结构的存在性。

链复形的同调极限与同调余极限

Homotopy Limits and Homotopy Colimits of Chain Complexes

本文通过cobar和bar构造给出链复形图的同调极限与余极限公式，并证明Bousfield-Kan公式在任何带框架模型范畴中计算同调余极限。

coCartesian fibrations and homotopy colimits

The main objective of this paper is to show that the homotopy colimit of a diagram of quasi-categories and indexed by a small category is a localization of Lurie's higher Grothendieck construction of the diagram. We thereby generalize Thomason's classical result which states that the homotopy colimit of a diagram of categories has the homotopy type of (the classifying space of) the Grothendieck construction of the diagram of categories.

Free duals and a new universal property for stable equivariant homotopy theory

We study the left adjoint $\mathbb{D}$ to the forgetful functor from the $\infty$-category of symmetric monoidal $\infty$-categories with duals and finite colimits to the $\infty$-category of symmetric monoidal $\infty$-categories with finite colimits, and related free constructions. The main result is that $\mathbb{D} \mathcal C$ always splits as the product of 3 factors, each characterized by a certain universal property. As an application, we show that, for any compact Lie group $G$, the $\infty$-category of genuine $G$-spectra is obtained from the $\infty$-category of Bredon (\emph{a.k.a} …

CAT中同伦极限与余极限的互换

Interchanging homotopy limits and colimits in CAT

设$I$，$J$是小范畴，$C:I\times J@>>>\CAT$是小范畴的函子.证明了若$I$有最终对象，则正则映射$\operatorname{hocolim}_J\operatorname{holim}_IC@>>>\operatorname{holim}_I \operatorname{hocolim}_JC$是强同伦等价.

The Partial Simplicial Category and Algebras for Monads

We construct explicitly the weights on the simplicial category so that the colimits and limits of 2-functors with those weights provide the Kleisli objects and the Eilenberg-Moore objects, respectively, in any 2-category.

内部范畴的余极限

Colimits of internal categories

研究在特定1-范畴$\mathcal{E}$条件下，内部范畴的2-范畴$\mathbf{Cat}(\mathcal{E})$的性质，证明其有有限2-余极限等，还给出了逆结果，核心方法是利用范畴相关性质及伴随关系进行推导。

Cauchy completeness for DG-categories

We go back to the roots of enriched category theory and study categories enriched in chain complexes; that is, we deal with differential graded categories (DG-categories for short). In particular, we recall weighted colimits and provide examples. We solve the 50 year old question of how to characterize Cauchy complete DG-categories in terms of existence of some specific finite absolute colimits. As well as the interactions between absolute weighted colimits, we also examine the total complex of a chain complex in a DG-category as a non-absolute weighted colimit.

建模空间的极限、余极限与谱

Limits, Colimits, and Spectra of Modelled Spaces

本文提出 $T_0$-模型谱的替代构造以重新证明 Coste 对偶伴随，并将其推广到相对谱，证明相关建模空间范畴存在极限与余极限。

Tight limits and completions from Dedekind-MacNeille to Lambek-Isbell

While any infimum in a poset can also be computed as a supremum, and vice versa, categorical limits and colimits do not always approximate each other. If I approach a point from below, and you approach it from above, then we will surely meet if we live in a poset, but we may miss each other in a category. Can we characterize the limits and the colimits that approximate each other, and guarantee that we will meet? Such limits and colimits are called *tight*. Some critically important network applications depend on them. This paper characterizes tight limits and colimits, and describes tight com…

On the construction of limits and colimits in $\infty$-categories

In previous work, we introduce an axiomatic framework within which to prove theorems about many varieties of infinite-dimensional categories simultaneously. In this paper, we establish criteria implying that an $\infty$-category - for instance, a quasi-category, a complete Segal space, or a Segal category - is complete and cocomplete, admitting limits and colimits indexed by any small simplicial set. Our strategy is to build (co)limits of diagrams indexed by a simplicial set inductively from (co)limits of restricted diagrams indexed by the pieces of its skeletal filtration. We show directly th…

The colimit of an $\infty$-local system as a twisted tensor product

We describe several equivalent models for the infinity-category of infinity-local systems of chain complexes over a space using the framework of quasi-categories. We prove that the given models are equivalent as infinity-categories by exploiting the relationship between the differential graded nerve functor and the cobar construction. We use one of these models to calculate the quasi-categorical colimit of an infinity-local system in terms of a twisted tensor product.

在交换范畴中重新审视极限和余极限的精确性

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。

泛同伦理论

Universal homotopy theories

给定一个小范畴C，我们证明了有一个普遍的方法将C扩展成一个模型范畴，本质上是通过形式上邻接同伦上极限。本地化技术成为一种将“关系”强加给这些通用小工具的方法。本文发展了这种形式主义，并讨论了各种应用，例如研究同伦colimits，德怀尔-坎理论的框架，并同伦理论的计划。

Limits and colimits of crossed groups

Although the notion of crossed groups was originally introduced only in the simplicial case, the definition makes sense in the other categories. For instance, Batanin and Markl studied crossed interval groups to investigate symmetries on the Hochschild cohomology in view of operads. The aim of this paper is to make a comprehensive understanding of crossed groups for arbitrary base categories. In particular, we focus on the local presentability of the category of crossed groups, monadicity, and the basechange theorem along certain sorts of functors. The paper also contains the classification of…

通过充实同伦理论的三范畴普遍性质

Tricategorical Universal Properties Via Enriched Homotopy Theory

本文通过充实于Gray模型范畴的同伦良好极限与余极限，建立了三范畴极限与余极限的理论，并将其应用于伪单子Kleisli构造、中心构造及双范畴严格化等具体结构。

Metric abstract elementary classes as accessible categories

We show that metric abstract elementary classes (mAECs) are, in the sense of [LR] (i.e. arXiv:1404.2528), coherent accessible categories with directed colimits, with concrete $\aleph_1$-directed colimits and concrete monomorphisms. More broadly, we define a notion of $κ$-concrete AEC---an AEC-like category in which only the $κ$-directed colimits need be concrete---and develop the theory of such categories, beginning with a category-theoretic analogue of Shelah's Presentation Theorem and a proof of the existence of an Ehrenfeucht-Mostowski functor in case the category is large. For mAECs in par…

双范畴中的单子与余单子范畴

On categories of monads and comonads in double categories

本文研究双范畴中单子范畴的幺半性、余完备性和局部可表现性，并推广到余单子情形，主要依赖于平行和稳定局部余极限等概念。

Enriched cofibration categories

Cofibration categories are a formalization of homotopy theory useful for dealing with homotopy colimits that exist on the level of models as colimits of cofibrant diagrams. In this paper, we deal with their enriched version. Our main result claims that the category $[\mathcal{C},\mathcal{M}]$ of enriched diagrams equipped with the projective structure inherits a structure of a cofibration category whenever $\mathcal{C}$ is locally cofibrant (or, more generally, locally flat).

关于自由幺半群构造的注记

Note on the construction of free monoids

在具有有限极限和可数上极限的幺半范畴范畴中构造自由幺半群，其中任意一侧的张量保持可数链的自反余均衡器和上极限。

Homotopy Colimits of DG Categories and Fukaya Categories

We construct a new cylinder object for semifree differential graded (dg) categories in the category of dg categories. Using this, we give a practical formula computing homotopy colimits of semifree dg categories. Combining it with the result of Ganatra, Pardon, and Shende, we get a formula computing wrapped Fukaya categories of Weinstein manifolds using their sectorial coverings. This formula has lots of applications including a practical computation of the wrapped Fukaya category of any cotangent bundle or plumbing space. In this paper, we compute wrapped Fukaya categories of cotangent bundle…

Traces in monoidal derivators, and homotopy colimits

A variant of the trace in a monoidal category is given in the setting of closed monoidal derivators, which is applicable to endomorphisms of fiberwise dualizable objects. Functoriality of this trace is established. As an application, an explicit formula for the trace of the homotopy colimit of endomorphisms over finite EI-categories is deduced. This result can be seen as a generalization of the additivity of traces in monoidal categories with a compatible triangulation.