Talk: Accessible and Presentable infinity categories
Disclaimer
- these are supplementary notes for my talk about "Accessible and Presentable infinity categories"
- I have tried to add more content (especially proofs I will skip)
- unfortunately my notes contain errors and are not self-sustained
Default Notation
mathematical
- \(\mu\) any cardinal
- \(\kappa\) typically a regular cardinal
- \(K\) a small simplicial set (for some inaccessible cardinal)
- \(X\) any simplicial set
- \(p: K \rightarrow \mathcal{C}\) some infinity functor
- \(\mathcal{C}, \mathcal{D}\) infinity categories
- \(\mu\)-blabla: omitting the \(\mu\) typically means, that it holds for some cardinal \(\mu\) (in our fixed Universe)
- \(\kappa\)-blablabla for a regular cardinal \(\kappa\): omitting the \(\kappa\) typically means \(\omega\)-blablabla (where \(\omega = \vert \mathbb{N} \vert\)
- We won't jump universes in anything I have prepared.
sources
- HTT: Higher Topos Theory, Jacob Lurie
- FW: \(\infty\)-Categories in Topology, Ferdinand Wagner (available at https://github.com/FlorianAdler/inftyCats)
Content of my talk
Plan
- Motivation
- some set theory
- introduce "smallness" notions & filteredness
- introduce accessible & presentable categories
- alternative characterization of Acc/Pres. Categories using Ind
- Given time: \(\mathrm{An}\) and \(\mathrm{Cat}_{\infty}\) are presentable
motivation - necessity of set theory
- we need some form of "smallness"
- we want some form of "locally smallness" for our yoneda embedding
- we want some form of "small (co-)limits" and (small) (co-)completeness:
- we want some form of "locally smallness" for our yoneda embedding
- set theory + limits \(\leadsto\) gives "some control over (co-)limits"
Cardinals
Size issues
definition
filtered
compact
- kappa infinity compact object
- motivation for regular:
- kappa filtered for a singular cardinal
- kappa compact functor - we might actually need larger regular cardinals
- kappa filtered for a singular cardinal
- kappa small colimit of kappa compact objects
- representable functor as kappa compact object
accessible & presentable
examples
1-cat
constructions
- infinity functor category into an accessible category as accessible category / infinity functor category into a presentable category as presentable category
- infinity slice category of an accessible category as accessible category / infinity slice category of a presentable category as presentable category
- infinity coslice category of an accessible caetgory as accessible category / infinity coslice category of a presentable category as presentable category