Notes on enriched categories with colimits of some class (completed version)
G. M. Kelly, V. Schmitt
备注：
This is a completion of CT/0501383. Results presented here are mainly from unpublished notes of the first author and contains those in CT/0309209 and CT/0403164
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} weights, the classes $\p^+$ and $\p^-$ both coincide with the class $\Q$ of {\em absolute} weights. For any class $Φ$ and any category $\A$, we have the free $Φ$-cocompletion $Φ(\A)$ of $\A$; and we recognize $\Q(\A)$ as the Cauchy-completion of $\A$. We study the equivalence between ${(\Q(\A^{op}))}^{op}$ and $\Q(\A)$, which we exhibit as the restriction of the Isbell adjunction between ${[\A,\V]}^{op}$ and $[\A^{op},\V]$ when $\A$ is small; and we give a new Morita theorem for any class $Φ$ containing $\Q$. We end with the study of $Φ$-continuous weights and their relation to the $Φ$-flat weights.
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We develop a general framework for studying signatures, presentations, and algebraic colimits of enriched monads for a subcategory of arities, even when the base of enrichment $\mathcal{V}$ is not locally presentable. When $\mathcal{V}$ satisfies the weaker requirement of local boundedness, the resulting framework is sufficiently general to apply to the $Φ$-accessible monads of Lack and Rosický and the $\mathcal{J}$-ary monads of the first author, while even without local boundedness our framework captures in full generality the presentations of strongly finitary monads of Lack and Kelly as we…
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…
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…
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…
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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.
This is the first of a series of papers on enriched infinity categories, seeking to reduce enriched higher category theory to the higher algebra of presentable infinity categories, which is better understood and can be approached via universal properties. In this paper, we introduce enriched presheaves on an enriched infinity category. We prove analogues of most familiar properties of presheaves. For example, we compute limits and colimits of presheaves, prove that all presheaves are colimits of representable presheaves, and prove a version of the Yoneda lemma.
We show that some recent constructions in the literature, named `weak' generalizations, can be systematically treated by passing from 2-categories to categories enriched in the Cartesian monoidal category of Cauchy complete categories.
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…
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 develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for simplicially-enriched categories, links homotopy locally presentable V-categories with combinatorial model V-categories, in the case where has all objects of V are cofibrant.
Sifted colimits (those that commute with finite products in sets) play a major role in categorical universal algebra. For example, varieties of (many-sorted) algebras are precisely the free cocompletions under sifted colimits of (many-sorted) Lawvere theories. Such a characterisation does not depend on the existence of finite products in algebraic theories, but on the above fact that these products commute with sifted colimits and another condition: finite products form a sound class of limits. In this paper we study the notion of soundness for general classes of weights in enriched category t…
Let $\mathcal C$ be a $\mathcal V$-enriched model category. We say that an object $x$ of $\mathcal C$ is homotopy tiny if the total right derived functor of $\mathcal C(x, -) : \mathcal{C} \rightarrow {\mathcal V}$ preserves homotopy weighted colimits. Let $\mathcal C_0$ be a full subcategory of $\mathcal C$ all of whose objects are homotopy tiny. Our main result says that the homotopy category of the category generated by $\mathcal C_0$ under weak equivalences and homotopy weighted colimits is equivalent to the homotopy category of the category $\mathcal V^{\mathcal C_0^{op}}$ of $\mathcal V$…
For our concepts of change of base and comonadicity, we work in the general context of the tricategory $\mathrm{Caten}$ whose objects are bicategories $\mathscr{V}$ and whose morphisms are categories enriched on two sides. For example, for any monoidal comonad $G$ on a cocomplete closed monoidal category $\mathscr{C}$, the forgetful functor $U : \mathscr{C}^G\to \mathscr{C}$ is comonadic when regarded as a morphism in $\mathrm{Caten}$ between one-object bicategories. We show that the forgetful pseudofunctor $\mathscr{U}:\mathscr{V}^\mathscr{G}\rightarrow \mathscr{V}$ from the bicategory of Eil…
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. …
This paper studies a general notion of flatness in the enriched context: P-flatness where the parameter P stands for a class of presheaves. One obtains a completion of a category A by considering the category Flat_P(A) of P-flat presheaves over A. This completion is related to the free cocompletion of A under a class of colimits defined by Kelly. For a category A, for P = P0 the class of all presheaves, Flat_P0(A) is the Cauchy-completion of A. Two classes P1 and P2 of general interest for general metric spaces are considered. The P1- and P2-flatness are investigated and the associated complet…
This paper studies a notion of parameterized flatness in the enriched context: p-flatness where the parameter p stands for a class of presheaves. One obtains a completion of a category A by considering the category F_p(A) of p-flat presheaves over A. The completion is related to the free cocompletion under a class of colimits defined by Kelly. We define a notion of Q-accessible categories where Q is the class of p-flat indexes. For a category A, for p = P0 the class of all presheaves, F_P0(A) is the Cauchy-completion of A. Two classes P1 and P2 of interest for general metric spaces and prorder…
We study 2-monads and their algebras using a Cat-enriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2-categorical points of view. Every 2-category with finite limits and colimits has a canonical model structure in which the weak equivalences are the equivalences; we use these to construct more interesting model structures on 2-categories, including a model structure on the 2-category of algebras for a 2-monad T, and a model structure on a 2-category of 2-monads on a fixed 2-category K.
