Tags
- #higher-algebra/derived #higher-algebra/infty-cats #resources #todo/learning
Refs:
- Unsorted/Introduction to infinity categories
- A crash course on infty cats #resources/videos
- Short notes: https://people.maths.ox.ac.uk/brantner/L4.pdf, https://www.imo.universite-paris-saclay.fr/~brantner/Spring22L2.pdf #resources/notes
- Seminar notes #resources/notes
- https://pi.math.cornell.edu/~dmehrle/notes/cornell/18fa/homotopy #resources/notes
- A short course on infty categories: https://people.math.rochester.edu/faculty/doug/otherpapers/groth_scinfinity.pdf#page=1 #projects/lecture-notes
- Recs from Glaeser
  - For treatments that cover quasi-categories in detail, there’s Cisinski’s book (https://cisinski.app.uni-regensburg.de/CatLR.pdf), and Rezk’s notes https://faculty.math.illinois.edu/~rezk/quasicats.pdf.
  - For “big-picture” treatments (which gloss over technical details in order to get quickly to some interesting applications) there’s
  - Mazel-Gee’s notes (https://etale.site/teaching/w21/math-128-lecture-notes.pdf) which @Reuben Stern (they/them) already mentioned, as well as
  - Groth’s short course on infinity-categories (https://arxiv.org/pdf/1007.2925.pdf).
Links:
- Kan extension
- simplicial set
- Kan complex
- cartesian fibration
- stable infinity category
- pregeometry
- infty topos
- presentable category
- localization (category theory)
- derived ring
- nerve
- modules over a category
- simplicial category
- higher category
- Kan complex
- Kan extension
- simplicial set
- stable infinity category
- infinity groupoids
- classifying space
- homotopy type
- Kan fibration
- Models:
  - quasicategory
  - Complete Segal spaces
  - Gamma space?
- skeleta
- hypercovering
- hyper-descent
- Waldhausen S construction for infinity categories

infinity categories

Misc

attachments/Pasted%20image%2020220425095015.png

What is an infinity category?

attachments/Pasted%20image%2020220318002310.png

How to build an infty category: attachments/Pasted%20image%2020220429235334.png

An \(\infty{\hbox{-}}\)category \(\mathcal{C}\) is a (large) simplicial set] \(\mathcal{C}\) such that any diagram of the form

Pasted image 20210515015420

admits the indicated lift, where \(\Lambda_i^n\) is an \(i{\hbox{-}}\)horn (a simplex missing the \(i\)th face) for \(0 < i < n\).

All inner horns are fillable, i.e. simplicial set are inner Kan complexes.
Different to Kan complexes, which include all \(i\).

Notes

∞-categories form a (large) ∞-category.
The Segal condition, essentially characterizes \(\infty{\hbox{-}}\)categories among simplicial infinity groupoids
Given two ∞-categories \(\mathsf{D}, \mathsf{C}\), there is a functor ∞-category \({\mathsf{Fun}}(\mathsf{D}, \mathsf{C})\).
- In terms of quasicategory, the hom here is internal hom in simplicial set.
- Example: for a given ∞-category \(\mathsf{I}\) we have the ∞-category of presheaves \({\mathsf{Fun}}(\mathsf{I}^{\operatorname{op}}, { \underset{\infty}{ {\mathsf{Grpd}}} })\)

-In practice, ∞-categories are constructed from existing ones by constructions that automatically guarantee that the result is again an ∞-category, - The construction typically uses universal properties in such a way that the resulting ∞-category is only defined up to equivalence - Can take a homotopy category

For each \(n \geq 0\) there is a cat \(\Delta[n] = { \mathcal{N}({\left\{{0 \leq 1 \leq \cdots \leq n}\right\}}) }\).
Commutative triangles in \(\mathsf{C}\): objects in the functor category \({\mathsf{Fun}}(\Delta[2], \mathsf{C})\)
\({ \underset{\infty}{ \mathsf{Cat}} }\leq {\mathsf{Kan}}\): infinity categories are a subcategory of Kan complexes.