Quasi-category

Summary

In mathematics, more specifically , a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a . The study of such generalizations is known as . Quasi-categories were introduced by . André Joyal has much advanced the study of quasi-categories showing that most of the usual basic and some of the advanced notions and theorems have their analogues for quasi-categories. An elaborate treatise of the theory of quasi-categories has been expounded by . Quasi-categories are certain simplicial sets. Like ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between these objects (1-simplices). But unlike categories, the composition of two morphisms need not be uniquely defined. All the morphisms that can serve as composition of two given morphisms are related to each other by higher order invertible morphisms (2-simplices thought of as "homotopies"). These higher order morphisms can also be composed, but again the composition is well-defined only up to still higher order invertible morphisms, etc. The idea of higher category theory (at least, higher category theory when higher morphisms are invertible) is that, as opposed to the standard notion of a category, there should be a mapping space (rather than a mapping set) between two objects. This suggests that a higher category should simply be a . The model of quasi-categories is, however, better suited to applications than that of topologically enriched categories, though it has been proved by Lurie that the two have natural model structures that are Quillen equivalent. By definition, a quasi-category C is a simplicial set satisfying the inner Kan conditions (also called weak Kan condition): every inner horn in C, namely a map of simplicial sets where , has a filler, that is, an extension to a map . (See Kan fibration#Definitions for a definition of the simplicial sets and .

Official source

https://en.wikipedia.org/wiki/Quasi-category

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Related publications (4)

Hopf algebras and Hopf-Galois extensions in infinity-categories

Aras Ergus

In this thesis, we study interactions between algebraic and coalgebraic structures in infinity-categories (more precisely, in the quasicategorical model of (infinity, 1)-categories). We define a notion of a Hopf algebra H in an E-2-monoidal infinity-category and lift some results about ordinary Hopf algebras, such as the fundamental theorem of Hopf modules, to this setting. We also study Hopf-Galois extensions in this context. Given a candidate Hopf-Galois extension, i.e., a map f : A -> B of H-comodule algebras where H coacts on A trivially, we construct a structured version of the comparison map B (x)_A B -> H (x) B that allows us to compare the category of descent data for f with a category of "B-modules equipped with a semilinear coaction of H". We provide further insights into the case of commutative (i.e., E-infinity) comodule algebras over a commutative Hopf algebra, for instance a description of the aforementioned category of modules equipped with a semilinear coaction as the limit of a "categorified cobar construction". Moreover, we provide a simple description of comodules over a space in slice categories of the infinity-category of spaces, which enables us to realize multiplicative Thom objects as comodule algebras and thus incorporate them into the aforementioned framework.

EPFL2022

Quasi-categories vs. Segal spaces: Cartesian edition

Nima Rasekh

We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: 1.On marked simplicial sets (due to Lurie [31]), 2.On bisimplicial spaces (due to deBrito [12]), 3.On bisimplicial sets, 4.On marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal–Tierney and the straightening construction due to Lurie.

2021

The cotangent complex and Thom spectra

Nima Rasekh

The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of E-infinity-ring spectra in various ways. In this work we first establish, in the context of infinity-categories and using Goodwillie's calculus of functors, that various definitions of the cotangent complex of a map of E-infinity-ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an E-infinity-ring spectrum and Pic(R) denote its Picard E8-group. Let M f denote the Thom E-infinity- R-algebra of a map of E-infinity-groups f : G. Pic(R); examples of M f are given by various flavors of cobordism spectra. We prove that the cotangent complex of R -> M f is equivalent to the smash product of M f and the connective spectrum associated to G.

2021

Related concepts (7)

Kan fibration

In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan. For each n ≥ 0, recall that the , , is the representable simplicial set Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard -simplex: the convex subspace of Rn+1 consisting of all points such that the coordinates are non-negative and sum to 1.

Quasi-category

Model category

In mathematics, particularly in homotopy theory, a model category is a with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes ( theory). The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic K-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results.

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