Daniel C. Isaksen
We present some constructions of limits and colimits in pro-categories. These are critical tools in several applications. In particular, certain technical arguments concerning strict pro-maps are essential for a theorem about étale homotopy types. Also, we show that cofiltered limits in pro-categories commute with finite colimits.
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We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene Theorem on regular languages.
We prove that a category which is symmetric (relaxed) monoidal closed, (small) complete, well-powered and has a small cogenerating family, is cocomplete.
Thomason's Homotopy Colimit Theorem has been extended to bicategories and this extension can be adapted, through the delooping principle, to a corresponding theorem for diagrams of monoidal categories. In this version, we show that the homotopy type of the diagram can be also represented by a genuine simplicial set nerve associated with it. This suggests the study of a homotopy colimit theorem, for diagrams $\b$ of braided monoidal categories, by means of a simplicial set {\em nerve of the diagram}. We prove that it is weak homotopy equivalent to the homotopy colimit of the diagram, of simplic…
In this thesis weighted colimits in 2-categories equipped with promorphisms are studied. Such colimits include most universal constructions with counits, like ordinary colimits in categories, weighted colimits in enriched categories, and left Kan extensions. In the first chapter we recall the notion of 2-categories equipped with promorphisms (also simply called equipments), that provide a coherent way of adding bimodule-like morphisms to a 2-category. In the second chapter we recall two ways of defining weighted colimits in equipments. Most important to us is their original definition, introdu…
Limits and colimits of synthetic $\infty$-categories
本文在合成 simplicial 同调类型理论框架下，研究了 ∞-范畴的极限与余极限理论，并展示了空间族的极限可通过依赖积计算。
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…
In a paper by Ford, it is claimed that to any pushout square of categories with all involved functors injective, there is associated an exact "Mayer--Vietoris" sequence of derived (co)limits. We provide a counter-example to this general statement. Further, we construct the Mayer--Vietoris sequence under some restrictions that cover the well-known case of a pushout square of group monomorphisms.
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…
Lenses are an important tool in applied category theory. While individual lenses have been widely used in applications, many of the mathematical properties of the corresponding categories of lenses have remained unknown. In this paper, we study the category of small categories and asymmetric delta lenses, and prove that it has several good exactness properties. These properties include the existence of certain limits and colimits, as well as so-called imported limits, such as imported products and imported pullbacks, which have arisen previously in applications. The category is also shown to b…
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。
A restriction category is an abstract formulation for a category of partial maps, defined in terms of certain specified idempotents called the restriction idempotents. All categories of partial maps are restriction categories; conversely, a restriction category is a category of partial maps if and only if the restriction idempotents split. Restriction categories facilitate reasoning about partial maps as they have a purely algebraic formulation. In this paper we consider colimits and limits in restriction categories. As the notion of restriction category is not self-dual, we should not expect …
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。
Given a class Phi of weights, we study the following classes: Phi^+ of Phi-flat weights which are the psi for which psi-colimits commute in the base V with limits with weights in Phi; and Phi^-, dually defined, of weights psi for which psi-limits commute in the base V with colimits with weights in Phi. We show that both these classes are saturated (i.e. closed under the terminology of Albert-Kelly or Betti's coverings). We prove that for the class P of all weights P^+ = P^-. For any small B, we defined an enriched adjunction a` la Isbell [B,V]^op -> [B^op,V] and show how it restricts to an equ…
The paper is in essence a survey of categories having $ϕ$-weighted colimits for all the weights $ϕ$ in some class $Φ$. We introduce the class $Φ^+$ of {\em $Φ$-flat} weights which are those $ψ$ for which $ψ$-colimits commute in the base $\V$ with limits having weights in $Φ$; and the class $Φ^-$ of {\em $Φ$-atomic} weights, which are those $ψ$ for which $ψ$-limits commute in the base $\V$ with colimits having weights in $Φ$. We show that both these classes are {\em saturated} (that is, what was called {\em closed} in the terminology of \cite{AK88}). We prove that for the class $\p$ of {\em all…
This document is centered around a main idea: simplicial categories, by which we mean simplicial objects in the category of categories, can be treated as a two-fold categorical structure and their double category theory is homotopically meaningful. The most well-known two-fold structures are double categories, typically used to organize bimodules in various contexts. However there is no double category of spaces even though notions of bimodule are conceivable. We first remedy this defect of double category theory by constructing a meaningful simplicial category of spaces. Then we develop the a…
We prove that the category of Banach $L^0$-modules over a given $σ$-finite measure space is both complete and cocomplete, which means that it admits all small limits and 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…
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.
The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically finitely accessible categories can be characterized as: (a) free cocompletions of small categories under filtered colimits, and (b) categories of flat presheaves on some small category. The equivalence between (a) and (b) is what makes the theory so general and fruitful. Notions of enriched accessibility have also been considered in the literature for various base…
We show that the homotopy colimit construction for diagrams of categories with an operad action, recently introduced by Fiedorowicz, Stelzer and Vogt, has the desired homotopy type for diagrams of weak braided monoidal categories. This provides a more flexible way to realize E-2 spaces categorically.
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.
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。
We consider categories of posets with $\mathfrak{C}$-valued structure sheaves for any category $\mathfrak{C}$ and see how they possess poset-indexed lax colimits that are both easy to describe and "weakly equivalent" to their ordinary colimits in a certain sense. We employ this construction to study descent problems on schematic spaces -- a particular scheme-like kind of ringed poset -- , proving a general Seifert-Van Kampen Theorem for their étale fundamental group that recovers and generalizes the homonym result for schemes to the topology of flat monomorphisms. The techniques are general en…
Free colimit completion in $\infty$-categories
本文研究 $\infty$-范畴中的自由余极限完备化，证明 Ind-构造的多项有用性质可推广至任意此类构造。
Characterising colimiting omega-cocones of projection pairs in terms of least upper bounds of their embeddings and projections is important to the solution of recursive domain equations. We present a universal characterisation of this local property as omega-cocontinuity of locally continuous functors. We present a straightforward proof using the enriched Yoneda embedding. The proof can be generalised to Cattani and Fiore's notion of locality for adjoint pairs.
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。
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.
We define and discuss lax and weighted colimits of diagrams in $\infty$-categories and show that the coCartesian fibration associated to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free Cartesian fibration associated to a a functor of $\infty$-categories. As an application of these results, we prove that lax representable functors are preserved under exponentiation, and also that the total space of a presentable Cartesian fibration between $\infty$-categories is presentable, generalizing a theorem of Makkai and Paré to t…
The higher algebra of weighted colimits
本文在弱双充实∞-范畴框架下建立加权余极限理论，证明存在性、余端表达、自由余完备化等结果，并构造了多种高阶范畴的张量积结构。
Colimits of categories, zig-zags and necklaces
本文提供了小范畴图表余极限的组合描述，引入锯齿形双范畴记录识别，并在单纯enriched背景下为Dugger和Spivak的项链定理提供了更简短的证明。
Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal in this paper is expository: we explain both approaches and a proof of their equivalence. Our second goal is to generalize this result to enriched categories and homotopy weighted limits, showing that the classical explicit constructions still give the right answer in the abstract sense. This result partially bridges the gap between classical homo…
In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L^2-Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of I-indexed categories where I is any small category admitting a finite I-CW-model for its I-classifying space. Special cases of our Homotopy Colimit Formula include formulas for products, homotopy pushouts, homotopy orbits, and transport groupoids. We also apply our formulas to Haefliger complexes of groups, which extend Bass--Serre graphs of groups to highe…
On the lack of colimits in various categories arising in pointfree topology and algebraic logic
研究证明McKinsey-Tarski代数范畴不等价于代数品种，反驳了Peter Jipsen的问题。各类BAO、Heyting代数及框架的范畴不完整，因此不等价于预品种。
We provide a complete classification of all the possible categoricity spectra, in terms of internal size, that can appear in a large accessible category with directed colimits, assuming the Singular Cardinal Hypothesis ($SCH$), and providing as well explicit threshold cardinals for eventual categoricity. This includes as a particular case the first complete classification of categoricity spectra of abstract elementary classes (AEC's) entirely in $ZFC$. More specifically, we have the following theorem: Let $\mathcal{K}$ be a large $κ$-accessible category with directed colimits. Assume the Singu…
We summarize our axioms for higher categories, and describe the blob complex. Fixing an n-category C, the blob complex associates a chain complex B_*(W;C)$ to any n-manifold W. The 0-th homology of this chain complex recovers the usual topological quantum field theory invariants of W. The higher homology groups should be viewed as generalizations of Hochschild homology (indeed, when W=S^1 they coincide). The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold W. We outline the important properties of the blob complex, and sketch the pro…
We provide, among other things: (i) a Bousfield--Kan formula for colimits in $\infty$-categories (generalizing the 1-categorical formula for a colimit as a coequalizer of maps between coproducts); (ii) $\infty$-categorical generalizations of Barwick--Kan's Theorem B$_n$ and Dwyer--Kan--Smith's Theorem C$_n$ (regarding homotopy pullbacks in the Thomason model structure, which themselves vastly generalize Quillen's Theorem B); and (iii) an articulation of the simultaneous and interwoven functoriality of colimits (or dually, of limits) for natural transformations and for pullback along maps of di…
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…
We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, if and only if homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left adjoints. These characterizations generalize to an abstract notion of "stability relative to a class of functors", which includes in particular pointedness, semiadditivity, and ordinary stability. To prove them, we develop the theory of derivators enriched over monoidal left derivators and weighted homotopy limits and colimits therein.
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。
We provide a proof, in $ZFC$, of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC's). Moreover, assuming in addition the Singular Cardinal Hypothesis ($SCH$), we prove a direct generalization to the more general context of accessible categories with directed colimits. If $\mathcal{K}$ is such a category, we show that there is a cardinal $μ$ such that if $\mathcal{K}$ is $λ$-categorical for some $λ\geq μ$ (i.e., it has only one object of internal size $λ$ up to isomorphism), then $\mathcal{K}$ is eventually categorical (i.e., it is $λ'$-categorical for every $λ' \g…
We generalize the constructions and results of Chapter 10 in Baldwin's "Categoricity" to coherent accessible categories with concrete directed colimits and concrete monomorphisms. In particular, we prove that if any category of this form is categorical in a successor, directed colimits of saturated objects are themselves saturated.
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。
Categorical Ambidexterity
本文证明了对于允许一组余极限的∞-范畴的双刃剑结果，统一并扩展了可呈现∞-范畴的极限和余极限识别以及∞-范畴的∞-半可加性。
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} …
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。
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.
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…
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).
In this paper we prove that various quasi-categories whose objects are $\infty$-categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a general theorem in which we characterize the data that is required to define a limit cone in a quasi-category constructed as a homotopy coherent nerve. Since all quasi-categories arise this way up to equivalence, this analysis covers the general case. Namely, we show that quasi-categorical limit cones may be modeled at the point-set level by pseudo homotopy limit…
A dagger category is a category equipped with a functorial way of reversing morphisms, i.e. a contravariant involutive identity-on-objects endofunctor. Dagger categories with additional structure have been studied under different names e.g. in categorical quantum mechanics and algebraic field theory. In this thesis we study the dagger in its own right and show how basic category theory adapts to dagger categories. We develop a notion of a dagger limit that we show is suitable in the following ways: it subsumes special cases known from the literature; dagger limits are unique up to unitary isom…
We introduce and study the Scott adjunction, relating accessible categories with directed colimits to topoi. Our focus is twofold, we study both its applications to formal model theory and its geometric interpretation. From the geometric point of view, we introduce the categorified Isbell duality, relating bounded (possibly large) ionads to topoi. The categorified Isbell duality interacts with the Scott adjunction offering a categorification of the Scott topology over a poset (hence the name). We show that the categorified Isbell duality is idempotent, similarly to its uncategorified version. …
The goal of this paper is to demystify the role played by the Reedy category axioms in homotopy theory. With no assumed prerequisites beyond a healthy appetite for category theoretic arguments, we present streamlined proofs of a number of useful technical results, which are well known to folklore but difficult to find in the literature. While the results presented here are not new, our approach to their proofs is somewhat novel. Specifically, we reduce the much of the hard work involved to simpler computations involving weighted colimits and Leibniz (pushout-product) constructions. The general…
In this paper we study the question of how to transfer homotopic structure from the category sD of simplicial objects in a fixed category D to D. To this end we use a sort of homotopy colimit s : sD --> D, which we call simple functor. For instance, the Bousfield-Kan homotopy colimit in a Quillen simplicial model category is an example of simple functor. As a remarkable example outside the setting of Quillen models we include Deligne simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also describe a natura…
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