Right, this kind of thing is indeed what I have in mind.

Another example (with Cory Knapp) is a lifting monad induced by a 
dominance. Fix a universe U. Then its type of propositions, Prop, lives in 
the next universe U'. A dominance is a subset of Prop subject to certain 
conditions. Prop itself is a dominance, and let's consider this for 
simplicity. Then a partial element of a type X is a proposition P (the 
extent of definition of the partial element) together with a function P->X. 
The lifting of X is then 

   LX := Sigma(P:U), isProp P * (P->X). 

If X is in a universe V, then LX is in the universe U' \/ V (namely the 
least universe after U' and V, where we are assuming a sequence of 
universes). However, if we apply L once more to get L(L X), this is in the 
same universe as L X (we increase the universe levels only once), and we 
get well typed functions eta : X->LX and mu : L(LX)->LX that satisfy the 
monad laws. 

If we assume propositional resizing, then all propositions live in the 
first universe U0, and so does Prop, and then L becomes a monad in the 
usual sense. But it is not clear what is gained by this (in this example) 
other than getting something one is more familiar with.  

In other examples, resizing does make a difference (of course). Consider 
for example the assertion that Prop is a complete lattice with respect to 
the "->" ordering . If we say that every family has a least upper bound, 
then we don't need resizing to prove that (we use the propositional 
truncation of the sum of the family to calculate the join). But to get  
that every *subset* of Prop (that is, map s : Prop->Prop) has a least upper 
bound, we would need resizing, as the natural candidate Sigma(P:Prop), s(P) 
is a proposition in the next universe and hence is not in Prop unless we 
have resizing. In this second example, the problem is solved by working 
with families rather than subsets. Are there examples in which there is no 
(known) way out without resizing?

Martin


On Wednesday, 24 January 2018 22:40:59 UTC, Nicola Gambino wrote:
>
> Dear Martin, 
>
> On 24 Jan 2018, at 22:36, Martín Hötzel Escardó <escar...@gmail.com 
> <javascript:>> wrote:
>
> So one vague question is how much one can do *without* propositional 
> resizing and living with the fact that universe levels may go up and down 
> in constructions such as the above. (A vague answer is "a lot", from my own 
> experience of formalizing things.) 
>
> A more precise question is that if we have a monad "up to universe 
> juggling" (such as the above), what kind of universal property "up to 
> universe juggling" does it correspond to. 
>
>
> You may have a look at relative monads (Altenkirch et al) and relative 
> pseudomonads (Fiore, Gambino, Hyland, Winskel). We considered the presheaf 
> construction that takes a small category to a locally small one (and hence 
> jumps up a universe) as a relative pseudomonad. Here, “pseudo” refers to 
> coherence issues, which I am not sure arise in type theory.
>
> Best wishes,
> Nicola
>
> Dr Nicola Gambino
> School of Mathematics, University of Leeds
> Web: http://www1.maths.leeds.ac.uk/~pmtng/ 
>
>