Notes on limits of accessible categories

Abstract

Let $\kappa$ be a regular cardinal, $\lambda<\kappa$ be a smaller infinite cardinal, and $\mathsf K$ be a $\kappa$-accessible category where colimits of $\lambda$-indexed chains exist. We show that various category-theoretic constructions applied to $\mathsf K$, such as the inserter and the equifier, produce $\kappa$-accessible categories $\mathsf E$ again, and the most obvious expected description of the full subcategory of $\kappa$-presentable objects in $\mathsf E$ in terms of $\kappa$-presentable objects in $\mathsf K$ holds true. In particular, if $\mathsf C$ is a $\kappa$-small category, then the category of functors $\mathsf C\rightarrow\mathsf K$ is $\kappa$-accessible, and its $\kappa$-presentable objects are precisely all the functors from $\mathsf C$ to the $\kappa$-presentable objects of $\mathsf K$. We proceed to discuss the preservation of $\kappa$-accessibility by conical pseudolimits, lax and oplax limits, and weighted pseudolimits. The results of this paper go back to an unpublished 1977 preprint of Ulmer. Our motivation comes from the theory of flat modules and flat quasi-coherent sheaves.