Infinity-categories reading group

Simplicial sets and simplicial objects, Peter Webb

talk notes

Definitions: the category Delta, simplicial sets and objects. Examples: ordered simplicial complexes (almost!), the standard simplices, the nerve of a category. Definition: (inner) horns, Kan complexes, infinity-categories.  

References: 

• Higher categories and homotopical algebra, Cisinski

• An elementary illustrated introduction to simplicial sets, Friedman

Introduction to infinity-categories, Tyler Lawson

talk notes

Infinity categories arise as a possible solution to three problems that come up quite naturally: (P1) what is a good, common framework that allows us to deal with things like categories up to equivalence, spaces up to homotopy equivalence, chain complexes up to chain homotopy equivalence, keeping track of all the data that starts to appear? (P2) how can we make sure that the constructions we might want to do in such a framework are equivalence-invariant (i.e. that we are free to replace an object by an equivalent one, and not mess up the construction)? and (P3) how can we deal with "homotopy (co)limits" in such a way that they still retain some version of a universal property? 

Categories enriched in spaces, Pranjal Dangwal

talk notes

Definition: infinity-groupoid of a space. (Informal) definitions of (n,k)-categories, with a discussion of the cases (1,1)---in connection to categories---, (infinity, 0)---in connection to the infinity-groupoid of a space---and (infinity, 1). Definition of enriched categories and examples, and special focus on enrichment in topological spaces (aka, compactly generated weakly hausdorff, the "nice ones"). Discussion about the notion of equivalence between categories enriched in spaces, and a few examples left to the reader.

References: Higher Topos Theory, Lurie (Chapters 1 and Appendix A.1)

Segal categories and Segal spaces, David DeMark

talk notes

Brief historical remarks on homotopy theory; basics on Quillen model categories, Quillen adjunctions, Quillen equivalences. Definition of (complete) Segal spaces, and connection with simplicial categories. Brief description of the model structure for complete Segal spaces. Definition: Segal (pre)categories; brief description of the model structure.

Relating different constructions, Jake Hinds

talk notes

After recalling the model structures on the categories of simplicial sets (whose fibrant objects are quasi-categories), on Segal pre-categories (whose fib obj are Segal categories) and on bisimplicial sets (whose fib obj are complete Segal spaces), this talk presented an adjunction between SeCat and CSS, and another adjunction between QCat and CSS. Evidence that these are Quillen adjunction was given by showing that the right adjoints preserve fibrant objects; moreover, these are Quillen equivalences, explaining why all these are alternative models for (infinity,1)-categories. 

Are simplicial categories scary or nice? Shuhang Xue

talk notes

Review of enriched categories, with focus on Top-enriched and sSet-enriched categories, and Quillen equivalences between these two categories. Main question is: what is the relation between these and quasi-categories? To answer this, we first asked: how can we get a version of the nerve for a simplicially enriched category, that captures the simplicial information? this led to the definition of a "thickening" of Delta[n], and of the simplicial nerve. One can then use this to get a version of the theorem that says a simplicial set has unique fillings for inner horns if and only if it's the nerve of a category! The new theorem states that if a simplicial category is locally Kan, then it's simplicial nerve is a quasi-category.  

References: Higher Topos Theory

Limits and colimits, Garrett Credi

Spectra and stable infinity-categories, Melissa Wei

talk notes

Introduction to stable infinity-categories, suspension and loop functors, and how the homotopy category is triangulated. Main example: the infinity-category of spectra, and its homotopy category, the "stable homotopy category". This was defined as the stabilization of Top, also called the category of spectrum objects in Top, which was introduced more generally for any pointed infinity-category.

References: 

• Higher Algebra, Lurie

• A short course in infinity-categories, Groth

• Spectra in model categories and applications to the algebraic cocomplex, Schwede

Monoidal structures, Connor Bass

talk notes

Review of symmetric monoidal categories, and in particular, how to encode them as (certain types of) functors from Fin_* to Cat. Review of (co)cartesian morphisms and (co)cartesian fibrations, and their connection to symmetric monoidal categories and their corresponding algebra objects. Overview of how this perspective can be generalized to the infinity-setting to produce a notion of symmetric monoidal (infinity,1)-category, and an (infinity,1)-category of (commutative) algebras.