nLab AccCat

Idea

By $AccCat$ (often just $Acc$ or $\mathbf{Acc}$ etc.) one typically means the (very large) 2-category whose

• objects are accessible categories,

• 1-morphisms are accessible functors

• 2-morphisms are natural transformations.

Hence forgetting the accessiblity conditions yields a (non-full) 2-functor from $AccCat$ to the 2-category Cat of all categories, functors and natural transformations.

Properties

2-Limits

Proposition

The 2-category AccCat has all (lax) 2-limits and these are preserved by the inclusion $AccCat \to$ Cat.

This appears as Makkai & Paré (1989), Thm. 5.1.6, Cor. 5.1.8, Adámek & Rosický (1994) around Thm. 2.77.

Proposition

Given a cosmos for enrichment $\mathcal{V}$ which is (symmetric monoidal closed and) locally presentable, then the 2-category $\mathcal{V}$-AccCat of $\mathcal{V}$-enriched accessible categories has all PIE 2-limits and splittings of idempotent equivalences, equivalently it has all flexible 2-limits as well as 2-pullbacks along isofibrations.

The analogous statements holds for $\mathcal{V}$-enriched and conically accessible categories, in which case the forgetful functor $\mathcal{V}\text{-}ConAccCat \to \mathcal{V}\text{-}Cat$ preserves these 2-limits.

This is Lack & Tendas (2023), Thm. 5.5, Thm. 5.9.

Proposition

(directed unions)
The 2-category AccCat has directed 2-colimits of systems of fully faithful functors. If there is a proper class of strongly compact cardinals, then it has directed colimits of systems of faithful functors.

Paré & Rosický (2013)

Related entries

• accessible category (see the respective section there)

• Cat

References

• Michael Makkai, Robert Paré, Accessible categories: The foundations of categorical model theory, Contemporary Mathematics 104. American Mathematical Society (1989) [ISBN:978-0-8218-7692-3]

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series 189, Cambridge University Press (1994) [doi:10.1017/CBO9780511600579]

• Robert Paré, Jiří Rosický, Colimits of accessible categories, Math. Proc. Cambr. Phil. Soc. 155 (2013) 47-50 [doi:10.1017/S0305004113000030, arXiv:1110.0767]

• Stephen Lack, Giacomo Tendas, Virtual concepts in the theory of accessible categories, Journal of Pure and Applied Algebra 227 2 (2023) 107196 [arXiv:10.1016/j.jpaa.2022.107196, arXiv:2205.11056]