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!

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    $\begingroup$ 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 $\endgroup$ – Denis-Charles Cisinski May 7 at 21:21
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    $\begingroup$ Model structures where the lifts are unique are "uninteresting". See here. $\endgroup$ – Zhen Lin May 8 at 1:29
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    $\begingroup$ @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. $\endgroup$ – Denis-Charles Cisinski May 8 at 12:53

As Denis-Charles said in a comment, if you require the unique lifting property on a weak factorization system, it becomes a "unique" or "orthogonal" factorization system. In the paper Bousfield localisation and colocalisation of one-dimensional model structures, Balchin and Garner studied model categories whose constituent weak factorization systems are unique factorization systems, in particular giving a number of interesting examples. (However, most "higher categorical" or "homotopical" model categories are not one-dimensional; this is a very special, one might even say degenerate, property.)

Having no more than one lift may be an interesting relation to study between classes of maps, but I doubt that it would give rise to a notion of "factorization system" or "model category" that bears much resemblance to the usual one.


If your goal is to understand how lifting properties in algebraic geometry fit into the bigger picture, I think the thing to say is that lifting properties are widely used in category theory for all sorts of purposes (historically, I wouldn't doubt that their usage was kicked off by Grothendieck). You might start with the nlab page on orthogonality. Unique vs. non-unique lifitng properties play slightly different roles; the "unique if exists" lifitng property as in unramifiedness is rarely used elsewhere.

To be slightly more precise, both unique and non-unique lifting properties are used not just to construct factorization systems and weak factorization systems, but also to give definitions of various categories as subcategories of other categories. This is why the orthogonal subcategory problem is important in category theory.


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