nLab AccCat




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

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




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

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


Given a cosmos for enrichment 𝒱\mathcal{V} which is (symmetric monoidal closed and) locally presentable, then the 2-category 𝒱\mathcal{V}-AccCat of 𝒱\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 𝒱\mathcal{V}-enriched and conically accessible categories, in which case the forgetful functor 𝒱-ConAccCat𝒱-Cat\mathcal{V}\text{-}ConAccCat \to \mathcal{V}\text{-}Cat preserves these 2-limits.

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


(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)


Created on May 3, 2023 at 08:19:46. See the history of this page for a list of all contributions to it.