Triposes, exact completions, and Hilbert's ε-operator

Abstract

Triposes were introduced as presentations of toposes by J.M.E. Hyland, P.T. Johnstone and A.M. Pitts. They introduced a construction that, from a tripos $P:\mathcal{C}^\mathrm{op} \rightarrow \mathbf{Pos}$, produces an elementary topos $\mathcal{T}_P$ in such a way that the fibration of the subobjects of the topos $\mathcal{T}_P$ is freely obtained from $P$. One can also construct the “smallest” elementary doctrine made of subobjects which fully extends $P$, more precisely the free full comprehensive doctrine with comprehensive diagonals $P_\mathrm{cx}:\mathcal{Prd}_P\,^\mathrm{op}\rightarrow \mathbf{Pos}$ on $P$. The base category has finite limits and embeds into the topos $\mathcal{T}_P$ via a functor $K:\mathcal{Prd}_P \rightarrow \mathcal{T}_P$ determined by the universal property of $P_\mathrm{cx}$ and which preserves finite limits. Hence it extends to an exact functor $K^\mathrm{ex}:(\mathcal{Prd}_{P})_\mathrm{ex/lex} \rightarrow \mathcal{T}_P$ from the exact completion of $\mathcal{Prd}_P$.

We characterize the triposes $P$ for which the functor $K^\mathrm{ex}$ is an equivalence as those $P$ equipped with a so-called $\varepsilon$-operator. We also show that the tripos-to-topos construction need not preserve $\varepsilon$-operators by producing counterexamples from localic triposes constructed from well-ordered sets.

A characterization of the tripos-to-topos construction as a completion to an exact category is instrumental for the results in the paper and we derived it as a consequence of a more general characterization of an exact completion related to Lawvere's hyperdoctrines.