We axiomatise the theory of $(\infty,n)$-categories. We prove that the space of theories of $(\infty,n)$-categories is a $B(\mathbb{Z}/2)^n$. We prove that Rezk's complete Segal $\Theta_n$-spaces, Simpson and Tamsamani's Segal $n$-categories, the first author's $n$-fold complete Segal spaces, Kan and the first author's $n$-relative categories, and complete Segal space objects in any model of $(\infty,n-1)$-categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of $(\mathbb{Z}/2)^n$.

On the Unicity of the Homotopy Theory of Higher Categories

The language of
 ∞-categories provides an insightful new way of expressing many
 results in higher-dimensional mathematics but can be challenging
 for the uninitiated. To explain what exactly an ∞-category is
 requires various technical models, raising the question of how
 they might be compared. To overcome this, a model-independent
 approach is desired, so that theorems proven with any model
 would apply to them all. This text develops the theory of
 ∞-categories from first principles in a model-independent
 fashion using the axiomatic framework of an ∞-cosmos, the
 universe in which ∞-categories live as objects. An ∞-cosmos is a
 fertile setting for the formal category theory of ∞-categories,
 and in this way the foundational proofs in ∞-category theory
 closely resemble the classical foundations of ordinary category
 theory. Equipped with exercises and appendices with background
 material, this first introduction is meant for students and
 researchers who have a strong foundation in classical 1-category
 theory.

Elements of ∞-Category Theory

We construct higher categories of iterated spans, possibly equipped with extra structure in the form of "local systems", and classify their fully dualizable objects By the Cobordism Hypothesis, these give rise to framed topological quantum field theories, which are the framed versions of the "classical" TQFTs considered in the quantization programme of Freed-Hopkins-Lurie-Teleman 
Using this machinery, we also construct an infinity-category of Lagrangian correspondences between symplectic derived algebraic stacks and show that all its objects are fully dualizable

Iterated spans and "classical" topological field theories

We introduce simple models for associative algebras and bimodules in the context of non-symmetric $\infty$-operads, and use these to construct an $(\infty,2)$-category of associative algebras, bimodules, and bimodule homomorphisms in a monoidal $\infty$-category. By working with $\infty$-operads over $\Delta^{n,\text{op}}$ we iterate these definitions and generalize our construction to get an $(\infty,n+1)$-category of $E_{n}$-algebras and iterated bimodules in an $E_{n}$-monoidal $\infty$-category. Moreover, we show that if $\mathcal{C}$ is an $E_{n+k}$-monoidal $\infty$-category then the $(\infty,n+1)$-category of $E_{n}$-algebras in $\mathcal{C}$ has a natural $E_{k}$-monoidal structure. We also identify the mapping $(\infty,n)$-categories between two $E_{n}$-algebras, which allows us to define interesting non-connective deloopings of the Brauer space of a commutative ring spectrum.

The higher Morita category of $E_n$-algebras

We introduce simple models for associative algebras and bimodules in the context of nonsymmetric ∞–operads, and use these to construct an (∞,2)–category of associative algebras, bimodules and bimodule homomorphisms in a monoidal ∞–category. By working with ∞–operads over Δn,op we iterate these definitions and generalize our construction to get an (∞,n+1)–category of En–algebras and iterated bimodules in an En–monoidal ∞–category. Moreover, we show that if C is an En+k–monoidal ∞–category then the (∞,n+1)–category of En–algebras in C has a natural Ek–monoidal structure. We also identify the mapping (∞,n)–categories between two En–algebras, which allows us to define interesting nonconnective deloopings of the Brauer space of a commutative ring spectrum.

https://msp.org/gt/2017/21-3/gt-v21-n3-p05-s.pdf

The higher Morita category of n–algebras

(∞, n)-Cat as a closed model category

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Clark Barwick

Author Tools

Chat about Author

Papers

(∞, n)-Cat as a closed model category