# Model categories with uniqueness

I've been learning about the construction of $$(\infty,1)$$-categories from simplicial sets, and more generally about the model category structure on simplicial sets, defined in terms of lifting properties w.r.t. horn inclusions etc.

My question is whether there is a sensible way to generalize the notion of a model category in terms of these right and left lifting properties, but where one can require lifts to be unique, or require that there be no more than one lift. If this exists, I would expect three possible relations between maps $$i$$ and $$p$$ (thinking of these as a horn inclusion and a generalized Kan fibration), namely:

• $$i$$ has the usual left lifting property w.r.t. $$p$$,
• $$i$$ has the unique lifting property w.r.t $$p$$: there exists a lift and it is unique,
• $$i$$ has no more than one lift with respect to $$p$$: there exists either no lift or one lift.

Has this been done before? It seems like a very natural thing to do, but I do not know whether it is possible or useful to generalize model categories likes this.

My two main motivations are:

• Defining ordinary categories as "fibrant" objects in simplicial sets, where the fibrations are now understood to be maps with the unique right lifting property w.r.t inner horn inclusions.
• Understanding valuative criteria in algebraic geometry. For example, if horn inclusions in simplicial sets are replaced by first order thickenings of affine schemes in the category of schemes, then morphisms of schemes with the unique right lifting property w.r.t. these thickenings are formaly étale morphisms, those with the not-necessarily-unique right lifting property are formally smooth, and those with the no-more-than-one-lift property are formally unramified.

If there are any references that could help me with this question, or some similar ideas that have been developed, I would like to hear about them!

• Maybe what you want is the general idea of a unique factorization system. There quite a few papers by Mathieu Anel: arXiv:0902.1130 on Grothendieck topologies in algebraic geometry, arXiv:2004.00731 on construction of localizations from unique factorization systems, which complements section 5.2.8 in Lurie's Higher Topos Theory. There are plenty of material to be found in n lab pages as well: ncatlab.org/nlab/show/factorization+system – Denis-Charles Cisinski May 7 at 21:21
• Model structures where the lifts are unique are "uninteresting". See here. – Zhen Lin May 8 at 1:29
• @Zhen Lin: Although I would recommend to read your posts very much for the good material in it, I would not qualify this sort of things as "uninteresting". $1$-categorical left Bousfield localizations exist in nature (e.g. sheaves, models of a Lawvere theory). Furthermore, this kind of concept is robust enough to survive $\infty$-categorical promotion, which gives even more important examples. This sounds interesting enough to me. – Denis-Charles Cisinski May 8 at 12:53