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Is there a reasonable definition for a category of categories equipped with a factorization system, say $({\bf C},(E,M))$ and functors $F\colon ({\bf C},(E,M))\to ({\bf D}, (E', M'))$ that preserve them? I see several possible choices, all unreasonable in some sense:

  1. $F(E)\subseteq E'$, and $F(M)\subseteq M'$; this is unreasonable, as it seems to strong $F=1_{\bf C}$ does not send $(All, Iso)$ to $(Iso, All)$. Do I have to interpret this as an analogue of the identity being non-continuous if the domain topology is coarser?
  2. $F$ preserves the left class. This is unreasonable, as it is not canonical. Why left, when $F$ is not a left adjoint?
  3. $F$ preserves the right class. This is unreasonable, as it is not canonical. Why right, when $F$ is not a right adjoint?
  4. $F(\mathbb F)\subseteq \mathbb{F}'$, if $\mathbb F =(E,M)$, and $\mathbb F' = (E', M')$. This means that $F(M)\subseteq M'$ and $F(E)\supseteq E'$.

I must admit I am confused.

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  • $\begingroup$ In 1., why do you expect to have such a map between cats with factorisation systems? Are you secretly thinking of model categories (or some variant thereof)? Why not have a notion of 'left map' and 'right map' for 2./3.? $\endgroup$
    – David Roberts
    Dec 25, 2016 at 23:49
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    $\begingroup$ You should have a double category that combines 2 and 3. (One is horizontal, the other vertical) $\endgroup$
    – Tim Campion
    Dec 26, 2016 at 0:50
  • $\begingroup$ Both are nice suggestions. I'll decide if they work tomorrow :-) $\endgroup$
    – fosco
    Dec 26, 2016 at 0:51
  • $\begingroup$ The particular case of your other question suggests analogy with defining morphisms of pairs $({\mathcal A}\subseteq{\mathbf C})$. Even in the case of pairs of sets, there are several possibilities like $A'\subseteq fA$, $fA\subseteq A'$, $f^{-1}A'\subseteq A$, $C'-A'\subseteq f(C-A)$ and versions with equalities. It depends on what do you want. Sort of a general abstract nonsense setup can be observed by viewing the objects as pairs $(C,A\in\wp\,C)$ where $\wp$ goes to categories and has several variances that may or may not be adjoint to each other. $\endgroup$ Dec 26, 2016 at 7:35
  • $\begingroup$ In other words, in that particular case you are looking at Grothendieck constructions of several co- and contravariant functors which happen to have coinciding values on objects, but entirely different values on morphisms - e. g. the power(po)set functor is in fact a trinity of functors, two of them covariant and one contravariant: $f^{-1}$ has both adjoints. $\endgroup$ Dec 26, 2016 at 7:45

1 Answer 1

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The main theorem of Korostenski and Tholen's Factorization systems as Eilenberg-Moore algebras proves that categories with (orthogonal) factorization systems are precisely the normal pseudo algebras for a 2-monad on Cat. The 2-monad in this case is the 2-monad that sends a category $C$ to its arrow category $C^2$ with the rest of the monad structure that you would guess (arising from the fact that the walking arrow category 2, like any category, is canonically a comonoid with respect to the cartesian product).

How do we think about this result? To equip a category $C$ with the structure of an algebra is to define a functor $C^2 \to C$ that is a (strict) retraction of the diagonal functor $C \to C^2$. (This is the meaning of "normal" above.) Some clever diagram chasing proves that such a functor equips $C$ with a functorial factorization, with the functor $C^2 \to C$ sending a morphism to the object through which it factors. To say that this data defines a pseudo-algebra for the 2-monad demands an additional associativity condition, up to natural isomorphism (hence the "pseudo"), and this can be seen to imply that the left factor of a right factor is an isomorphism and that the right factor of a left factor is an isomorphism. These conditions, together with functoriality, suffice to show that the functorial factorization defines an orthogonal factorization system.

Note also that for any 2-monad $T$ on Cat, there is a related 2-monad $T'$ so that strict $T'$-algebras are exactly pseudo $T$-algebras.

So in particular, categories with a factorization system are also the strict algebras for a 2-monad, if you would prefer.

2-monad theory now provides a general framework within which to answer Fosco's question. In general, there are four natural kinds of morphisms for algebras for a 2-monad --- strict, pseudo, lax, and oplax --- with the strict ones (preserving everything on the nose) only rarely encountered in the wild.

Suppose $C$ and $D$ are categories with factorization systems whose functorial factorizations are given by a pair of functors $Q : C^2 \to C$ and $E \colon D^2 \to D$ that send a morphism to the object through which it factors. A lax morphism of algebras specifies a functor $F \colon C \to D$ together with a natural transformation $EF \to FQ$ whose components define a commuting "comparison morphism" between the two factorizations of a map in the image of $F$. The presence of such a comparison implies that $F$ preserves the right classes.

Dually, a colax morphism specifies a functor $F \colon C \to D$ together with a natural transformation $FQ \to EF$ the presence of which implies that $F$ preserves the left classes. By taking mates, one can see that if $F \dashv U$, then $F$ defines a colax morphism of algebras if and only if $U$ defines a lax morphism of algebras, which explains why lax morphism structure feels natural for right adjoints and colax morphism structure feels natural for left adjoints.

Some preliminary remarks along these lines in the more general setting of algebraic weak factorization systems can be found in my thesis or (in a much more satisfying account) in Bourke and Garner's Algebraic weak factorization systems I.

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  • $\begingroup$ That's definitely the kind of answer I was hoping for. $\endgroup$
    – fosco
    Dec 27, 2016 at 21:13
  • $\begingroup$ I might need some references for the pseudo algebras part. I remember the Kan seminar related paper by lack on codescent objects proving something similar to Ps-T-Alg ~ T'-Alg for some T', but I'm not sure that's the reference you have in mind. Thanks again!! $\endgroup$
    – fosco
    Dec 28, 2016 at 1:42
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    $\begingroup$ Just to continue to harp on about double categories, according to the nlab page, the lax and colax morphisms between algebras for a 2-monad always fit together into a double category. $\endgroup$
    – Tim Campion
    Dec 28, 2016 at 18:00
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    $\begingroup$ Good answer. I just wanted to add that I think it's a little clearer to denote the arrow category $\rightarrow$ instead of $2$. As a general rule, I tend to think it's better to think of each number as denoting the corresponding discrete category, so that the familiar-looking equations like $2 \times \mathbf{C} \cong \mathbf{C} \sqcup \mathbf{C}$ and $\mathbf{C}^2 = \mathbf{C} \times \mathbf{C}$ can be true. Just my two cents. In any event, thanks for a very informative answer. $\endgroup$ Sep 13, 2017 at 6:05

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