An homotopical description of small presheaves (original) (raw)
An Algebraic Definition of (∞, N)-Categories
2015
In this paper we define a sequence of monads T(∞,n)(n ∈ N) on the category ∞-Gr of ∞-graphs. We conjecture that algebras for T(∞,0), which are defined in a purely algebraic setting, are models of∞-groupoids. More generally, we conjecture that T(∞,n)-algebras are models for (∞, n)-categories. We prove that our (∞, 0)-categories are bigroupoids when truncated at level 2. Introduction The notion of weak (∞, n)-category can be made precise in many ways depending on our approach to higher categories. Intuitively this is a weak∞-category such that all its cells of dimension greater than n are equivalences. Models of weak (∞, 1)-categories (case n = 1) are diverse: for example there are the quasicategories studied by Joyal and Tierney (see [24]), but also there are other models which have been studied like the Segal categories, the complete Segal spaces, the simplicial categories, the topological categories, the relative categories, and there are known to be equivalent (a survey of models ...
The stable category of preorders in a pretopos I: General theory
Journal of Pure and Applied Algebra, 2021
In a recent article Facchini and Finocchiaro considered a natural pretorsion theory in the category of preordered sets inducing a corresponding stable category. In the present work we propose an alternative construction of the stable category of the category PreOrd(C) of internal preorders in any coherent category C, that enlightens the categorical nature of this notion. When C is a pretopos we prove that the quotient functor from the category of internal preorders to the associated stable category preserves finite coproducts. Furthermore, we identify a wide class of pretoposes, including all σ-pretoposes and all elementary toposes, with the property that this functor sends any short Z-exact sequences in PreOrd(C) (where Z is a suitable ideal of trivial morphisms) to a short exact sequence in the stable category. These properties will play a fundamental role in proving the universal property of the stable category, that will be the subject of a second article on this topic.
The stable category of preorders in a pretopos II: the universal property
Annali di Matematica Pura ed Applicata (1923 -)
We prove that the stable category associated with the category PreOrd(C) of internal preorders in a pretopos C satisfies a universal property. The canonical functor from PreOrd(C) to the stable category Stab(C) universally transforms a pretorsion theory in PreOrd(C) into a classical torsion theory in the pointed category Stab(C). This also gives a categorical insight into the construction of the stable category first considered by Facchini and Finocchiaro in the special case when C is the category of sets.
Groupoids and skeletal categories form a pretorsion theory in Cat
We describe a pretorsion theory in the category Cat of small categories: the torsion objects are the groupoids, while the torsion-free objects are the skeletal categories, i.e., those categories in which every isomorphism is an automorphism. We infer these results from two unexpected properties of coequalizers in Cat that identify pairs of objects: they are faithful and reflect isomorphisms. * Partially supported by Ministero dell'Istruzione, dell'Università e della Ricerca (Progetto di ricerca di rilevante interesse nazionale "Categories, Algebras: Ring-Theoretical and Homological Approaches (CARTHA)). † The fourth author acknowledges partial financial assistance by Natural Sciences and Engineering Council of Canada under the Discovery Grants Program, no. 501260.
2023
The primary goal of this paper is to provide a comprehensive presentation of the fundamental groupoid, along with the necessary category theory required to define it and prove its fundamental properties. This will lead to the proof of the Seifert Van Kampen theorem for fundamental groupoids. From there, we will discuss basic notions of 2-category theory: this will allow us to explore the categorical properties of the fundamental groupoid when viewed as a costack over the category of 2-groupoids. We will show that, for a “nice" class of topological space, the fundamental groupoid is a terminal object in this category: this provides a purely categorical description of the fundamental groupoid. The focal point of this project is the discussion of this result, originally formulated and proved by Ilia Pirashvili in 2015. It serves as the main subject of investigation, offering valuable insights that will inform our concluding remarks and motivate questions for further research.
A Whirlwind Tour of the World of (infty,1)( infty,1)(infty,1)-categories
2013
This introduction to higher category theory is intended to a give the reader an intuition for what (infty,1)(\infty,1)(infty,1)-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from ordinary category theory generalize to this new setting, and what uses people have put the theory to. It is a rough guide to a vast terrain, focuses on ideas and motivation, omits almost all proofs and technical details, and provides many references.
Category Theory I: A gentle prologue
2024
[March 28, 2024 version] An update to the early chapters of my earlier Gentle Introduction notes, requiring only modest mathematical background. There are chapters on categories, and on constructions like products, pullbacks, exponentials that can occur in different categories. There is also a first encounter with functors. Part II continues the story, talking about natural transformations between functors, the Yoneda lemma and adjunctions and we take an introductory look at the idea of an elementary topos This book Part I is available as a cheap print-on-demand paperback from mid April, but I think of it as a beta version, still work in progress and all comments are still most welcome.
Homotopy Theory in Groupoid Enriched Categories
2005
The concepts of h-limits, strong h-limits (and their duals) and partial proofs of homotopy limit reduction theorems relating to h-limits and strong h-limits are already known for a groupoid enriched category (g.e. category). In this paper the concepts of weak h-limits, quasi-limits (and their duals) are introduced in a g.e. category and the fuller version of the homotopy limit reduction theorems concerning the four types of limits, i.e., weak h-limits, h-limits, strong h-limits and quasi-limits are proved. The previously called Brown Complement Theorem is proved under the restricted assumption that the g.e. category admits only weak h-limits instead of h-limits and the generalized version of the Brown Complement Theorem is also proved which is relevant to the problem of showing under suitable smallness conditions that if a g.e. category admits all h-limits then it also admits all h-colomits. 1. Introduction. In [3] Fantham and Moore have discussed the technique and language of category theory for doing homotopy theory. They have presented a reasonable approach for the category-theoretic aspects of homotopy theory via an enriched category. They have also proposed some concepts that arise from spaces, maps, and homotopy classes of homotopies of maps. As it stands, they comprise a special type of 2-category [4] in
2022
This dissertation is concerned with the foundations of homotopy theory following the ideas of the manuscripts Les Dérivateurs and Pursuing Stacks of Grothendieck. In particular, we discuss how the formalism of derivators allows us to think about homotopy types intrinsically, or, even as a primitive concept for mathematics, for which sets are a particular case. We show how category theory is naturally extended to homotopical algebra, understood here as the formalism of derivators. Then, we proof in details a theorem of Heller and Cisinski, characterizing the category of homotopy types with a suitable universal property in the language of derivators, which extends the Yoneda universal property of the category of sets with respect to the cocomplete categories. From this result, we propose a synthetic re-definition of the category of homotopy types. This establishes a mathematical conceptual explanation for the the links between homotopy type theory, ∞-categories and homotopical algebra, and also for the recent program of re-foundations of mathematics via homotopy type theory envisioned by Voevodsky. In this sense, the research on foundations of homotopy theory reflects in a discussion about the re-foundations of mathematics. We also expose the theory of Grothendieck-Maltsiniotis ∞-groupoids and the famous Homotopy Hypothesis conjectured by Grothendieck, which affirms the (homotopical) equivalence between spaces and ∞-groupoids. This conjectured, if proved, provides a strictly algebraic picture of spaces.
Left exact presheaves on a small pretopos
Journal of Pure and Applied Algebra, 1999
Given a small exact category A with finite colimits, we prove that the category Lex(R) of left exact presheaves on A is exact precisely when in X, the equivalence relation generated by
Pretorsion theories in general categories
Journal of Pure and Applied Algebra, 2021
We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair (T , F) of full replete subcategories in a category C, the corresponding full subcategory Z = T ∩F of trivial objects in C. The morphisms which factor through Z are called Z-trivial, and these form an ideal of morphisms, with respect to which one can define Zprekernels, Z-precokernels, and short Z-preexact sequences. This naturally leads to the notion of pretorsion theory, which is the object of study of this article, and includes the classical one in the abelian context when Z is reduced to the 0-object of C. We study the basic properties of pretorsion theories, and examine some new examples in the category of all endomappings of finite sets and in the category of preordered sets.
A Note on “The Homotopy Category is a Homotopy Category”
Journal of Mathematics and Statistics, 2019
In his paper with the title, "The Homotopy Category is a Homotopy Category", Arne Strøm shows that the category Top of topological spaces satisfies the axioms of an abstract homotopy category in the sense of Quillen. In this study, we show by examples that Quillen's model structure on Top fails to capture some of the subtleties of classical homotopy theory and also, we show that the whole of classical homotopy theory cannot be retrieved from the axiomatic approach of Quillen. Thus, we show that model category is an incomplete model of classical homotopy theory.
2016
In [1], Grothendieck develops the theory of pro-objects over a category C. The fundamental property of the category Pro(C) is that there is an embedding C c −→ Pro(C), the category Pro(C) is closed under small cofiltered limits, and these limits are free in the sense that for any category E closed under small cofiltered limits, pre-composition with c determines an equivalence of categories Cat(Pro(C), E)+ ≃ Cat(C, E), (where the "+" indicates the full subcategory of the functors preserving cofiltered limits). In this paper we develop a 2-dimensional theory of pro-objects. Given a 2-category C, we define the 2-category 2-Pro(C) whose objects we call 2-pro-objects. We prove that 2-Pro(C) has all the expected basic properties adequately relativized to the 2-categorical setting, including the universal property corresponding to the one described above. We have at hand the results of Cat-enriched category theory, but our theory goes beyond the Cat-enriched case since we consider the non strict notion of pseudo-limit, which is usually that of practical interest.
A theory of 2-pro-objects (with expanded proofs)
arXiv (Cornell University), 2014
In [1], Grothendieck develops the theory of pro-objects over a category C. The fundamental property of the category Pro(C) is that there is an embedding C c −→ Pro(C), the category Pro(C) is closed under small cofiltered limits, and these limits are free in the sense that for any category E closed under small cofiltered limits, pre-composition with c determines an equivalence of categories Cat(Pro(C), E)+ ≃ Cat(C, E), (where the "+" indicates the full subcategory of the functors preserving cofiltered limits). In this paper we develop a 2-dimensional theory of pro-objects. Given a 2-category C, we define the 2-category 2-Pro(C) whose objects we call 2-pro-objects. We prove that 2-Pro(C) has all the expected basic properties adequately relativized to the 2-categorical setting, including the universal property corresponding to the one described above. We have at hand the results of Cat-enriched category theory, but our theory goes beyond the Cat-enriched case since we consider the non strict notion of pseudo-limit, which is usually that of practical interest.
Toposes and Homotopy Toposes (Version 0.15)
2010
Contents 1. Grothendieck topos 1 2. The descent properties of a topos 8 3. Grothendieck topologies 4. Model Categories 5. Universal model categories and presentable model categories 6. Model topos 7. Truncation 8. Connectivity 9. The topos of discrete objects and homotopy groups 10. t-completion 11. Construction of model toposes References This document was created while I gave a series of lectures on "higher topos theory" in Fall 2005. At that time the basic references were the papers of Toen-Vezzosi [TV05] and a document of Lurie [Lura]; Lurie's book on higher topos theory was not yet available. The lectures were my attempt to synthesize what was known at the time (or at least, what was known to me). I've revised them in small ways since they were written. (For instance, I replaced the original term "patching" with the word "descent", as suggested by Lurie.) There are still some gaps in the exposition here.
Brief notes on category theory
2012
1. Every morphism f is associated with two objects (which may be the same) called the domain and codomain of f. One can view a morphism as an arrow from one object to another thus forming a directed graph. We sometimes write cod(f) and dom(f) to denote these objects, more often we give them names like A and B. We write f : A −→ B or A f −−→ B. ... 2. Given A f −−→ B and B g −−→ C we say f and g are composable (ie cod(f) = dom(g)). We define a binary operation, written ◦, on composable maps so that g ◦ f is a morphism: dom(g ◦ f) = dom(f) and cod(g ◦ f) = ...
Quasitopoi over a base category
2006
In this paper we develop the theory of quasispaces (for a Grothendieck topology) and of concrete quasitopoi, over a suitable base category. We introduce the notion of f-regular category and of f-regular functor. The f-regular categories are regular categories in which every family with a common codomain can be factorized into a strict epimorphic family followed by a (single) monomorphism. The f-regular functors are (essentially) functors that preserve finite strict monomorphic and arbitrary strict epimorphic families. These two concepts furnish the context to develop the constructions of the theory of concrete quasitopoi over a suitable base category, which is a theory of pointed quasitopoi. Our results on quasispaces and quasitopoi, or closely related ones, were already established by Penon, but we prove them here with different assumptions and generality, and under a completely different light.