Filtered colimits in the effective topos (Q819805)

The effective topos \({\mathcal E}ff\) is defined in the article of \textit{J. M. E. Hyland} [``The effective topos'', in: The L. E. J. Brouwer Centen. Symp., Proc. Conf., Noordwijkerhout/ Holl. 1981, Stud. Logic Found. Math. 110, 165--216 (1982; Zbl 0522.03055)]. Roughly speaking, the objects of this topos are effective sets (i.e., pairs \((X_0,=_X)\), where \(X\) is a set and \(=_X\) is a function from \(X^2\) to \(P(\mathbb{N})\), which denotes symmetric and transitive relation on \(X\)) and morphisms are maps between effective sets, which preserve this relation \(=_X\). This definition is given in details in the thesis of \textit{G. Rosolini} [``Continuity and effectiveness in topos'', Ph. D. Thesis, University of Oxford (1986)]. The category \({\mathcal E}ff\) is not a Grothendieck topos. The global sections functor \(\Gamma : {\mathcal E}ff \rightarrow {\mathcal S}ets\) has a right adjoint \(\delta : {\mathcal S}ets \rightarrow {\mathcal E}ff\). This note contains two theorems. First: Let \(\lambda > \omega\) be a regular cardinal. Then the following two assertions are equivalent: (i) The full subcategory of \({\mathcal E}ff\) in the \(\lambda\)-small projectives is dense. (ii) \(\delta\) preserves \(\lambda\)-filtered colimits. For \(\lambda = \omega\), the implication \(\text{(i)}\Longrightarrow \text{(ii)}\) still holds. Second: The functor \(\delta\) does not preserve \(\omega_1\)-filtered colimits.