Higher topos theory Lurie, Jacob; Lurie, Jacob
2009., 20090706, 2009, 2009-07-06, Volume:
170
eBook
Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher ...morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics.
We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal ∞-category. Our theory of enriched ∞-categories has many desirable properties; for instance, if the ...enriching ∞-category V is presentably symmetric monoidal then Cat∞V is as well. These features render the theory useful even when an ∞-category of enriched ∞-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and (∞,n)-categories). This is analogous to the advantages of ∞-categories over more rigid models such as simplicial categories — for example, the resulting ∞-categories of functors between enriched ∞-categories automatically have the correct homotopy type.
We construct the homotopy theory of V-enriched ∞-categories as a certain full subcategory of the ∞-category of “many-object associative algebras” in V. The latter are defined using a non-symmetric version of Lurie's ∞-operads, and we develop the basics of this theory, closely following Lurie's treatment of symmetric ∞-operads. While we may regard these “many-object” algebras as enriched ∞-categories, we show that it is precisely the full subcategory of “complete” objects (in the sense of Rezk, i.e. those whose spaces of objects are equivalent to their spaces of equivalences) that are local with respect to the class of fully faithful and essentially surjective functors. We also consider an alternative model of enriched ∞-categories as certain presheaves of spaces satisfying analogues of the “Segal condition” for Rezk's Segal spaces. Lastly, we present some applications of our theory, most notably the identification of associative algebras in V as a coreflective subcategory of pointed V-enriched ∞-categories as well as a proof of a strong version of the Baez–Dolan stabilization hypothesis.
This volume contains the proceedings of the Fourth Arolla Conference on Algebraic Topology, which took place in Arolla, Switzerland, from August 20-25, 2012. The papers in this volume cover topics ...such as category theory and homological algebra, functor homology, algebraic K -theory, cobordism categories, group theory, generalized cohomology theories and multiplicative structures, the theory of iterated loop spaces, Smith-Toda complexes, and topological modular forms.
We show that the integral cohomology rings of the moduli spaces of stable rational marked curves are Koszul. This answers an open question of Manin. Using the machinery of Koszul spaces developed by ...Berglund, we compute the rational homotopy Lie algebras of those spaces, and obtain some estimates for Betti numbers of their free loop spaces in case of torsion coefficients. We also prove and conjecture some generalisations of our main result.
This volume contains the proceedings of the WIT: Women in Topology workshop, held from August 18-23, 2013, at the Banff International Research Station, Banff, Alberta, Canada. The Women in Topology ...workshop was devoted primarily to active collaboration by teams of five to seven participants, each including senior and junior researchers, as well as graduate students.This volume contains papers based on the results obtained by team projects in homotopy theory, including $A$-infinity structures, equivariant homotopy theory, functor calculus, model categories, orbispaces, and topological Hochschild homology.
The clusters considered in this paper are seen as morphisms between small arbitrary diagrams in a given locally small category C. They have initially been introduced to extend to all small diagrams ...the results for filtered diagrams, by exhibiting a very basic presentation of the formula used in the definition of the category Ind(C) of ind-objects in C. They constitute a category Clu (C) which contains Ind(C). We study these clusters, their construction and composition. Thus we provide any user with the means to generate clusters and perform calculations with them. So we can give a simple proof of the fact that Clu (C) is a strict free cocompletion of C for all small diagrams, determined up to isomorphism. We compare it to some other cocompletion problems.
The formal theory of relative monads Arkor, Nathanael; McDermott, Dylan
Journal of pure and applied algebra,
September 2024, 2024-09-00, Volume:
228, Issue:
9
Journal Article
Peer reviewed
Open access
We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative comonads and relative ...coadjunctions follows by duality. While some aspects of the theory behave analogously to the non-relative setting, others require new insights. In particular, the universal properties that define the algebra object and the opalgebra object for a monad in a virtual equipment are stronger than the classical notions of algebra object and opalgebra object for a monad in a 2-category. Inter alia, we prove a number of representation theorems for relative monads, establishing the unity of several concepts in the literature, including the devices of Walters, the j-monads of Diers, and the relative monads of Altenkirch, Chapman, and Uustalu. A motivating setting is the virtual equipment ▪ of categories enriched in a monoidal category ▪, though many of our results are new even for ▪.
The aim of this article is to provide references to cognitive scientists, who are interested in learning category theory and using it in their research. This article consists of the three sections, ...question-and-answers on category theory, utility of category theory on cognitive science, and tutorial materials. In the question-and-answers on category theory, we answered to questions, with which beginners of category theory may come up. In the utility of category theory on cognitive science, we raised the three items of utility of category theory in building cognitive models. The learning materials share the books, slides, and videos on the web, recommended to start with.
Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs ...between strong (∞,n)-functors. We construct a double (∞,n)-category built out of the target (∞,n)-category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of Ed-algebras in a symmetric monoidal (∞,n)-category C to an (∞,n+d)-category using the higher morphisms in C.
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