TITLE: A locally presentable locally cartesian closed category that is not a quasitopos
QUESTION [14 upvotes]: This question asks for a locally presentable locally cartesian closed category that is not a topos.  All the answers given (at least in the 1-categorical case) are quasitoposes.  What is an example of a locally presentable locally cartesian closed 1-category that is not a quasitopos?
Note that there is no difference for this question whether "quasitopos" is meant in the elementary or the Grothendieck sense.  By C2.2.13 in Sketches of an Elephant, a locally presentable locally cartesian closed category is a Grothendieck quasitopos iff it is an elementary quasitopos iff it is quasi-effective, i.e. strong equivalence relations are effective.  So the question could equivalently be: what is an example of a locally presentable locally cartesian closed category that is not quasi-effective?

REPLY [8 votes]: Thomas Holder points out to me that this question is answered in Borceux and Pedicchio, A characterization of quasi-toposes, with an example that is reproduced in C4.2.4 of Sketches of an Elephant: the ind-completion of the category $\mathrm{Set}_c$ of finite or countably infinite sets.  It is locally finitely presentable essentially by definition, and it is locally cartesian closed because it is a "local exponential ideal" in the presheaf category of $\mathrm{Set}_c$.  But it is not a quasitopos because (1) it is balanced, so if it were a quasitopos it would be a topos, and (2) the image of $\mathbb{N}\in \mathrm{Set}_c$ satisfies Freyd's characterization of a natural numbers object, so if it lived in a topos it would be a NNO, but in a cocomplete topos an NNO cannot be finitely presentable.