Notes On Quasi-Categories: An Extension Of Category Theory.pdf

The notion of a quasi-category was introduced by Boardman and Vogt, generalizing the concept of category to include weaker structures.

## Elementary Aspects

- Universes: We define three Grothendieck universes: U1 (small), U2 (large), and U3 (extra-large). Entities in U1 are small, U2 is for large entities, and U3 for extra-large ones. Categories are locally small if their hom sets are small.
- Categories: We use standard notation for categories (Cat) and locally small large categories (CAT). Cat is large, and CAT is extra-large.
- Simplicial Sets: S = [∆o, Set] denotes the category of simplicial sets, with ∆ the category of finite non-empty ordinals and order-preserving maps. The simplicial interval I = ∆[1].
- Nerve Functor: N : Cat → S is the nerve functor, full and faithful, mapping a small category to its simplicial set nerve.
- Fundamental Category: τ1 : S → Cat is the left adjoint to N, preserving finite products. It assigns the fundamental category (homotopy category) to a simplicial set.

## Quasi-Categories

A quasi-category is defined as a simplicial set X where every inner horn Λk[n] → X has a filler ∆[n] → X. This generalizes the concept of Kan complex and nerve of a category.

### Examples and Properties:

- Kan Complexes: A simplicial set is a Kan complex if all horns have fillers.
- Nerve of Categories: The nerve of a category is a quasi-category, with unique fillers for inner horns.
- Large Quasi-Categories: Quasi-categories can be large, extending the concept to larger universes.
- Fundamental Category Description: The fundamental category hoX of a quasi-category X has a simpler description using homotopy classes of arrows and 2-simplices as homotopies.

## Further Topics

The paper covers various aspects of quasi-categories, including:

- Equivalence with simplicial categories
- Left and right coverings
- Join and slice operations
- Functors and their initial/final forms
- Morita equivalence
- Homotopy factorization systems
- Grothendieck fibrations
- Proper and smooth maps
- Localization
- Adjoint maps
- Cylinders, distributors, and spans
- Limits and colimits
- Kan extensions
- Span and duality
- The quasi-category Hot
- The trace
- Factorisation systems in quasi-categories
- Quasi-algebra
- Categories within quasi-categories
- Absolutely exact quasi-categories
- Descent theory
- Stable quasi-categories
- ∞-topos
- Higher quasi-categories
- Theta-categories