Exactly. In my note, I consider three universes (all fibrant) sProp sSet bSet sProp sub-presheaf of sSet sub-presheaf bSet sub-presheaf U They correspond to 3 properties G |- A sprop G |- A sset G |- A bset that can be described semantically in a simple way. For sprop and sset, to have a fibration structure is a property and G |- A sset and fibrant should correspond to the notion of covering space. But sSet does not correspond to a decidable type system, while it should be the case that sProp and bSet corresponds to a decidable type system. At least with bSet we validate axiom K and all developments that have been done using this axiom. ________________________________ From: Vladimir Voevodsky Sent: Monday, February 27, 2017 7:53 PM To: Thierry Coquand Cc: Prof. Vladimir Voevodsky; univalent-...@googlegroups.com; homotopytypetheory Subject: Re: [UniMath] [HoTT] about the HTS BTW, even if the universe of strict sets is not fibrant we can still have a judgement that something is a strict set and the rule that a = b implies a is definitionally equal to b if a and b are elements of a strict set. It is such a structure that would make many things very convenient. It is non- clear to, however, why typing would be decidable in such a system. Vladimir. On Feb 27, 2017, at 1:50 PM, Vladimir Voevodsky > wrote: On Feb 25, 2017, at 2:19 PM, Thierry Coquand > wrote: "Bishop set" which corresponds to the fact that any two paths between the same end points are -judgmentally- equal. This is not what I mean by a strict set. A strict set is a Bishop set where any two points connected by a path are judgmentally equal.