Twoples is a mentorship program in which a graduate student (me) is paired with an undergraduate student (Sergio).
∞-categories stuff:
- The construction of the ∞-category "underlying" a model category. (See [HTT, A.2], but check other sources since some places do it differently.) Not every ∞-category arises in this way -- what can we say about the ones that do? When can one compute eg. limits/colimits in the ∞-category from the model category?
- The "homotopy theory of ∞-categories" can be studied in many ways -- eg. by forming an ∞-category of ∞-categories, or a model category of ∞-categories. In particular, the first thing one needs to work out is what a "weak equivalence" of ∞-categories should be. Then one can show that a Quillen equivalence of model categories forms a weak equivalence of underlying ∞-categories.
- ...
NOTES:
RESOURCES:
- ∞-categories:
- "An Introduction to ∞-Categories" -- Moritz Groth
- Kerodon
- "Higher Topos Theory" -- Lurie
- "Higher Algebra" -- Lurie
- "Elements of ∞-category theory" -- Riehl and Verity
- "An invitation to higher algebra" -- Mazel-Gee
- Model categories:
- Dwyer-Spalinski
- "Homotopical Algebra" -- Quillen
- "Model Categories" -- Hovey
- "A Handbook on Model Categories" -- Balchin
- "Model Categories and Their Localizations" -- Hirschhorn
- Homological algebra:
- Homotopy limits/colimits:
- 2-categories:
- etc:
PROJECT IDEAS:
- Models of n-categories: In the Leinster paper, we saw tons of various models for various types of (weak) n-categories. But he doesn't go deeply into any of them.
- Comparing different models becomes difficult, because you kind of need to settle on a definition of an n-category to compare models of n-categories. This is because a category of n-categories will form an (n+1)-category (like how the category Cat forms a 2-category).
- In particular, take the n=2 case, and compare some of the models of them. I think Leinster already discusses what they look like for each of his models. This might be beyond the scope of this, but it would be interesting to just settle on one model for 3-categories and try to form a 3-category "2-Cat" of 2-categories, and describe what it means for two 2-categories to be "equivalent as 2-categories" (this should be some type of "weak equivalence" in the 3-category... whatever that means).
- Leinster doesn't talk about any EXAMPLES, so that would be nice. Especially examples of 2-categories shouldn't be hard to find (Cat is one). I think he does include some sources for applications, so that would be good to look into.
- What do limits look like in general n-categories (or 2-categories)? Kan extensions? Universal properties?
- The relation between weak n-categories and (n,n)-categories. Also see my question below about varying degrees of weakness in an (m.n)-category, depending on n.
- Something about ∞-categories..
- Comparing ∞-categories and model categories. So-called "underlying" ∞-categories. Showing that a Quillen equivalence of model categories induce an equivalence (whatever the right notion of equivalence here means) of underlying ∞-categories.
- Simplicial ("Dwyer-Kan") localization: In a series of 3 papers in 1980 (see the references here -- I also mention it somewhere in my thesis), Dwyer and Kan described a process of taking a category with weak equivalences and constructing out of it a simplicial category (simplicially enriched category). This basically just formally sticks in all homotopical/simplicial data by upgrading hom-sets to hom-simplicial sets, in a way that formally weakly inverts the weak equivalences. This is important because it's one of the early models for ∞-categories. It might be worth reading these carefully (some of their results have also been re-done in more modern language -- I think the nlab page has more on this).
- Homotopy limits and colimits, and how ∞-categories let you describe them somewhat naturally.
- SIZE CONDITIONS: Most people just choose to ignore these, but it would be good to actually dive into what these things mean. When do we consider small ∞-categories vs. large ones? Here might be a good starting place. These are related to Grothendieck universes, accessible cardinals, and all that...
QUESTIONS:
- What does it mean to say that homotopical algebra is a "non-abelian/non-linear" generalization of homological algebra -- how can we recognize classical homological algebra as a special type of homotopical algebra?
- What is the role of derived functors in the above, and how do Ext and Tor functors "measure non-abelianness/non-exactness" of a category?
- Model categories and infinity-categories are two ways of describing homotopy theories; what is the relationship between the two? How much can one transfer from one to the other? What are pros/cons to either?
- What is a homotopy limit/colimit? How do we describe them via model categories? Via ∞-categories? Can we relate them? How do they look specifically in the case of a (model or ∞-) category of spaces?
- What are Kan extensions, and how and why do they show up seemingly everywhere? (In particular, we can describe ordinary limits/colimits and homotopy limits/colimits as Kan extensions.) The ∞-version of Kan extensions are called (at least on n-lab) "homotopy Kan extensions" -- how do these work and relate to ordinary Kan extensions?
- The right notion of an equivalence of homotopy theories (described as model categories) is a "Quillen equivalence". What is the analogue in ∞-land, and how do they relate? (see this paper)
- Given a (strict? weak?) 2-category C, can we describe a "2-nerve" -- ie. a quasi-category NC whose objects, 1-morphisms, and 2-morphisms are those of C, with higher simplices witnessing compositions of 1-morphisms in a suitable way. (The homotopy 2-category of this guy as in [Riehl-Verity] should recover the 2-category C.) There are two ways to compose 2-morphisms in a 2-category: "horizontally" and "vertically" (see Cats in Context -- Riehl). Can we describe either or both of these in terms of horn filling in the "2-nerve"?
- There is something called a dg-category : a category that has hom-differential graded Z-modules. I think one can take these as a model for higher categories or ∞-categories? Or one can try... I think they're also related to A_{infinity} categories, but idk what that is.
- Read this paper about "universal homotopy theories" -- starting from a (small) category, one can form a model category by formally sticking in homotopy colimits.
- Learn something about 2-categories (I think there are some intro papers by Benabou, but there may be more modern sources). In particular, it seems that we can describe the ∞-versions of many categorical concepts (adjunctions, limits, ...) only up to the 2-level. How far does this go? I don't think all of ordinary category theory is 2-categorical, but it seems that a large part is (a hint of this is the Riehl-Verity stuff about the homotopy 2-category). Are there certain things that definitely cannot be described 2-categorically? See this talk (this is a great talk for just ∞-stuff in general, but I think he also mentions at some point the 2-categorical nature of some of the stuff).
- There are other models of homotopy theories besides ∞-categories and model categories. For example, "relative categories", which are like pared down model categories that have just the weak equivalences identified (there's this paper). There are also simplicial categories (which form a model of ∞-categories). Are some of these suited for certain things, or not suited for certain things?
- There is something called the Reedy model structure, which is a way of describing a model structure on a category of functors (ie. a category of the form Fun(C,D)). These show up for example when describing limits/colimits as adjoints (the functor categories in question look like Fun(I,C), ie. I-shaped diagrams in a category C). See here, but I'm sure there are better sources.
- Are (strict/weak) 2-categories or bi-categories the same as (∞,2)-categories, (2,1)-categories, (2,2)-categories, ... ?
- What do Kan extensions look like in 2-categories?
- The category Cat of (small) categories forms a 2-category (1-morphisms are functors and 2-morphisms are natural transformations). What about the category of 2-categories? Do they form a 3-category or something?
- I'm not sure if this is correct, but (2,2)-categories are weak 2-categories, (2,1)-categories are strict 2-categories, right? Is there a similar notion for varying weaknesses of 3-categories? Like (3,3)-cats should be weak 3-cats, (3,2)-cats should be "semi-strict" 3-cats, and (3,1)-cats should be strict 3-cats? More generally, there should be various (m,n)-categories for each n=1, 2, ..., m, describing varying degrees of "weakness" in an m-category.