nLab sound doctrine of limits

Idea

A class $\Phi$ of diagram shapes for limits — or more generally a class of weights for limits — is called sound as a doctrine of limits [Adámek, Borceux, Lack & Rosicky (2002)] if it behaves nicely when paired with the class $\Phi^+$ of all colimit shapes (or weights) that commute with $\Phi$-limits in Set (or more generally in the base of enrichment).

Definition

A collection of small categories $\mathbb{D}$ is a doctrine if, seen as a full subcategory of $Cat$, it is essentially small.

A doctrine is said to be sound if for every small category $C$ the following are equivalent.

1. For every $D \in \mathcal{D}$ and any functor $D^{op} \to C$ the corresponding category of cocones is connected.

2. $C$-colimits commute with $D$-limits in $Set$ for all $D \in \mathbb{D}$.

Note that we always have $2. \implies 1.$, [Adámek, Borceux, Lack, Rosicky, Prop.2.1]. A category $C$ satisfying 2. is called $\mathbb{D}$-filtered.

References

• Jiří Adámek, Francis Borceux, Stephen Lack, Jiri Rosicky, A classification of accessible categories, Journal of Pure and Applied Algebra 175 1–3 (2002) 7-30 [doi:10.1016/S0022-4049(02)00126-3]

• n-Cafe discussion about the above paper

• C. Centazzo, J. Rosický, E. M. Vitale, A characterization of locally D-presentable categories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 45 (2004), no. 2, pp. 141-146. numdam.

• Stephen Lack, Jiri Rosicky, Notions of Lawvere theory, arxiv

• Matěj Dostál, Jiří Velebil, An elementary characterisation of sifted weights, arxiv

• G. M. Kelly, V. Schmitt, Notes on enriched categories with colimits of some class, arXiv