Related papers: An Introduction to Higher Categorical Algebra

In this short survey we give a non-technical introduction to some main ideas of the theory of $\infty$-categories, hopefully facilitating the digestion of the foundational work of Joyal and Lurie. Besides the basic $\infty$-categorical…

We adapt the classical framework of algebraic theories to work in the setting of (infinity,1)-categories developed by Joyal and Lurie. This gives a suitable approach for describing highly structured objects from homotopy theory. A central…

The article is developing homological algebra in modules over non-unital rings and algebras. The main application is the definition and study of (directed) homology of $(\infty,1)$-categories and of directed spaces, including relative…

We discuss what it means for a symmetric monoidal category to be a module over a commutative semiring category. Each of the categories of (1) cartesian monoidal categories, (2) semiadditive categories, and (3) connective spectra can be…

We show that the $\infty$-category of normed algebras in genuine $G$-spectra, as introduced by Bachmann-Hoyois, is modelled by strictly commutative algebras in $G$-symmetric spectra for any finite group $G$. We moreover provide an analogous…

This chapter, written for "Stable categories and structured ring spectra," edited by Andrew J. Blumberg, Teena Gerhardt, and Michael A. Hill, surveys the history of homotopical categories, from Gabriel and Zisman's categories of fractions…

The notion of conformal algebras was introduced by Victor G. Kac using the axiomatic description of the operator product expansion of chiral fields in conformal field theory. The structure theory, representations and cohomology of Lie and…

Modern categories of spectra such as that of Elmendorf et al equipped with strictly symmetric monoidal smash products allows the introduction of symmetric monoids providing a new way to study highly coherent commutative ring spectra. These…

By homotopy linear algebra we mean the study of linear functors between slices of the $\infty$-category of $\infty$-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices…

In this review we give a detailed introduction to the theory of (curved) $L_\infty$-algebras and $L_\infty$-morphisms. In particular, we recall the notion of (curved) Maurer-Cartan elements, their equivalence classes and the twisting…

Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal $\infty$-categorical models for separable $C^*$-algebras $\mathtt{SC^*_\infty}$ and noncommutative spectra $\mathtt{NSp}$ using the framework of…

This is an introduction to the study of abstract homotopy theory by means of model categories and $(\infty,1)$-categories. The only prerequisites are very basic general topology and abstract algebra. None categorical background is needed.…

We define closed model category structures on different categories connected to the world of operad algebras over the category C(k) of (unbounded) complexes of k-modules: on the category of operads, on the category of algebras over a fixed…

In the present paper the cyclic homology functor from the category of $A_\infty$-algebras over any commutative unital ring $K$ to the category of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy…

In this paper we describe the homotopy category of the $A_\infty$categories. To do that we introduce the notion of semi-free $A_\infty$category, which plays the role of standard cofibration. Moreover, we define the non unital $A_\infty$…

Invited contribution to the Encyclopedia of Mathematical Physics. We give an introduction to the homotopical theory of higher categories, focused on motivating the definitions of the basic objects, namely $\infty$-categories and…

In paper arXiv:1406.1744, we constructed a symmetric monoidal category $LIE^{MC}$ whose objects are shifted (and filtered) L-infinity algebras. Here, we fix a cooperad $C$ and show that algebras over the operad $Cobar(C)$ naturally form a…

We study cohomology theories of strongly homotopy algebras, namely $A_\infty, C_\infty$ and $L_\infty$-algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of $C_\infty$-algebras thus generalising previous work…

We construct a left semi-model structure on the category of intensional type theories (precisely, on $\mathrm{CxlCat_{Id,1,\Sigma(,\Pi_{ext})}}$). This presents an $\infty$-category of such type theories; we show moreover that there is an…

Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page.…