Basic constructions on presentable categories

Theorem 22.2.2 (Basic constructions on presentable categories, [Lurie (2009), Theorem 5.5.1.1 and Propositions 5.5.3.6, 5.5.3.10--5.5.3.11 and 5.5.4.15]). The following statements hold.

(1)

For every small \(\infty \)-category \(C\), the presheaf category \(\PSh (C)\) is presentable. In particular, \(\An \simeq \PSh (*)\) is presentable.

(2)

If \(C\) is presentable and \(K\) is small, then the functor category \(\Fun (K,C)\) is presentable.

(3)

If \(C\) is presentable and \(X\in C\), then the slice categories \(C_{/X}\) and \(C_{X/}\) are presentable.

(4)

If \(C\) is presentable and \(S\) is a small collection of morphisms in \(C\), then the full subcategory \(C^{S\text {-}\mathrm {loc}}\) is presentable and its inclusion into \(C\) admits a left adjoint.