nLab exact (infinity,1)-category

1. Idea

Exact $(\infty,1)$-categories are the analog in (∞,1)-category theory of exact categories in category theory.

2. Definition

Definition 2.1. Let $\mathcal{C}$ be an (∞,1)-category. This is called an exact $(\infty,1)$-category if

1. $\mathcal{C}$ has a terminal object and (∞,1)-fiber products;

2. groupoid objects in $\mathcal{C}$ are effective:

3. realization of groupoid objects is universal.

3. Other notions of “exact”

There is another meaning for “exact (∞,1)-category” for which there is a Quillen Q-construction for exact (∞,1)-categories which allows to compute its algebraic K-theory.

4. Related concepts

• exact category

• finitely complete (infinity,1)-category

• regular (infinity,1)-category, coherent (infinity,1)-category?, (infinity,1)-pretopos

• Quillen Q-construction

• theorem of the heart

5. References

On exact $\infty$-categories

• Clark Barwick, On the Q construction for exact quasicategories (arXiv:1301.4725)

and the theorem of the heart:

• Clark Barwick, On exact infinity-categories and the Theorem of the Heart, Compositio Mathematica 151 (2015) 2160-2186 [arXiv:1212.5232, doi:10.1112/S0010437X15007447]