The n-Category Café

Maybe one place nForum doesn’t function as well as the Café is in linking discussions to older discussions. E.g., from the post we get sent here, but that doesn’t lead us back to an earlier discussion.

Maybe there’s little there, but you never know.

Toby has proposed a natural generalization of this to $k$-ary factorization systems.

The important thing about that is that it works even when $k$ is infinite. The generalisation to $k \gt 3$ but finite is due to Mike earlier in the thread.

If anybody is having trouble reconciling my infinitary version and Mike's finitary version, here is what you should notice: my $\alpha$ is Mike's $k + 1$, and while Mike's $k$-ary factorisation system has $k - 1$ (hence $\alpha - 2$) binary factorisation systems, my $\alpha$-stage factorisation system has $\alpha$ (hence $k + 1$) binary factorisation systems. In this case, my two extra factorisation systems are trivial ones of the form $(\text{all},\text{iso})$ and $(\text{iso},\text{all})$. (In general, these appear whenever $\alpha$ is bounded as a poset.)

Concerning ternary factorization and model categories:

might it be useful to think of this with “model 2-categories” in mind?

maybe replacing both the $(cof,triv-fib)$ binary system of model categories and $(triv-cof,fib)$-sytem each by a ternary system leads to some structure on a 2-category that can present $(\infty,2)$-categories?

(I have no real clue, just a guess, given that ternary factorization systems want to live in 2-categories, not 1-categories.)

Miles Gould and I came across $k$-ary factorization systems during the course of his PhD investigations, though I’m not sure that they actually made it into his thesis.

We had a leading example similar to yours, together with a variant, which I’ll now explain. (Maybe you’ve also discussed this at the nLab/Forum.)

The standard ‘BOFF’ factorization system on $Cat$ — bijective on objects/full and faithful — is really a factorization system on the category $Gph$ of directed graphs, lifted to $Cat$. There’s an easy theorem:

Theorem  Let $(E, M)$ be a factorization system on a category $C$. Let $T$ be a monad on $C$, and write $U: C^T \to C$ for the forgetful functor. Define $$\overline{E} = \{ f in C^T: U f \in E \}, \quad \overline{M} = \{ f in C^T: U f \in M \}.$$ Suppose that $T$ preserves $E$-arrows. Then $(\overline{E}, \overline{M})$ is a factorization system on $C^T$.

(You can find a proof as Lemma 3.0.7 of Miles’s thesis, though it’s not due to him, as he notes.)

For example, let $T$ be the free category monad on $Gph$. Take the BOFF factorization system on $Gph$. The image under $T$ of a bijective-on-objects map of graphs is again bijective on objects, so there is an induced factorization system on $Gph^T = Cat$ — namely, BOFF.

Now, there is a similar BOFF-like $(n + 1)$-ary factorization system on the category $n\text{-}Gph$ of $n$-graphs ($n$-globular sets). This is the ‘variant’ I mentioned. One might hope for an analogue of the theorem above, which would apply to the free strict $n$-category monad $T$ on $n\text{-}Gph$ to give the $(n + 1)$-factorization system on $n\text{-}Cat$.

I don’t think Miles and I thought very hard about what that analogous theorem should be. Perhaps someone else knows?

It might be useful to invoke Eugenia Cheng’s work on iterated distributive laws, which John discussed in This Week’s Finds 257.

A monad in the bicategory of spans is exactly a category. Bob Rosebrugh and Richard Wood showed some time ago that a distributive law between such monads is exactly a strict factorization system. By ‘strict’ I mean that every map factorizes uniquely as a map from the left class followed by a map from the right class. (Of course this is much too restrictive for most situations, but let’s go with it.)

A strict $k$-fold factorization system should therefore be defined as a $k$-fold distributive law. This means a sequence of $k$ monads on the same object of the bicategory of spans — that is, a sequence of $k$ categories with the same object-set — together with a distributive law of the $i$th over the $j$th whenever $i < j$, all subject to a Yang–Baxter equation for each $i < j < k$.

The definition is, of course, set up so that the usual facts about distributive laws extend to the $k$-fold setting. In particular, a $k$-fold distributive law tells you how to glue together those $k$ categories (monads in Span) to obtain a single category (monad in Span). This is the category on which the $k$-fold factorization system is defined.

What remains, then, is to move from the strict to the not-so-strict setting. Lack and Street’s ‘Formal theory of monads 2’ may be some help here.

Double factorisation systems are defined in:

A. Pultr, W. Tholen
Free Quillen Factorization Systems
Georgian Math. J.9 (2002), No. 4, 807-820

Re ternary factorisation systems

I studied ternary factorisation systems (and factorisation systems as commutation laws) with Joyal in 2003, but no publication ever came out of it.

Here is some input to the discussion, notably a notion of ambifibration, and some remarks on the importance of flatness when passing from strict to non-strict notions of factorisation system.

A naive ternary factorisation system on a category $C$ consists of three subcategories, $C_1$, $C_2$, $C_3$ each containing all the isomorphisms, such that every arrow factors uniquely (up to unique comparison isos) into $f_1 f_2 f_3$ with $f_i \in C_i$.

Proposition: Such a naive factorisation system is functorial (in the sense of Korostenski-Tholen) iff it is split (in the sense described by Mike in the original post).

This result provides some justification that this notion is a reasonable notion of ternary factorisation system.

The motivation for studying ternary factorisation systems was our notion of ambifibration, which I think may be of interest to this thread. A (Grothendieck) fibration is an example of a factorisation system, namely (vertical,cartesian), and this factorisation system lies over the trivial factorisation system in the base (id,all). Similarly, an opfibration is a factorisation system (opcartesian,vertical) lying over the trivial factorisation (all,id). By definition, an ambifibration (relative to a factorisation system $(A,B)$ in the base $C$) is a functor $p: X \to C$ such that every arrow in $A$ has an opcartesian lift, and every arrow in $B$ has a cartesian lift.

It follows that the restriction of $p$ to $A$ is an opfibration, and that the restriction of $p$ to $B$ is a fibration. Let $E$ denote the opcartesian arrows for the first restriction, and let $M$ denote the cartesian arrows for the second restriction. Now it is not difficult to prove: $(E,V,M)$ is a ternary factorisation system. (Here $V$ denotes the category of vertical arrows for $p$.) Conversely, given a ternary factorisation system $X = (E,V,M)$ and a functor to some category $C = (A,B)$ such that the restriction to $L = (E,V)$ is an opfibration over $A$, and the restriction to $R = (V,M)$ is a fibration over $B$, then the whole thing is an ambifibration.

I believe many of the ternary factorisation systems mentioned in this thread are in fact ambifibrations lying over some binary factorisation system.

(The reason for introducing the notion of ambifibration was that the ‘fat Delta’ is an ambifibration over $\Delta$. The fat Delta (introduced in my paper Weak identity arrows in higher categories) is a version of $\Delta$ where the degeneracy maps are only ‘up to homotopy’ in some sense. More precisely, if you visualise the objects of $\Delta$ as columns of dots (and functions as noncrossing strings), then fat Delta has as objects columns of dots where adjacent dots may or may not have a linking line. The morphisms are noncrossing strings that are injective on dots. This means that instead of the basic degeneracy map in $\Delta$ that merges two dots into one dot, there is in the fat Delta only the up-to-homotopy version where two non-linked dots map to two linked dots. The projection to $\Delta$ contracts the links. The fat Delta has a canonical ternary factorisation system lying over the the epi-mono factorisation system in $\Delta$. The middle class of arrows in the fat Delta are then the maps that lie over the identity maps in $\Delta$, so they are the ‘equivalences’. Often the middle class in the ternary factorisation system coming from an ambifibration has the flavour of equivalence, and in particular it will always satisfy 3-for-2. The Gabriel-Zisman localisation of the fat Delta along those vertical map gives back Delta again. Toen conjectured that also the Dwyer-Kan localisation of the fat Delta with respect to the vertical maps is equivalent to $\Delta$. This would provide some strong evidence for the correctness of the fat Delta as a combinatorial building block for a theory of higher categories with only weak identity arrows, which was my motivation for introducing it.)

Just as there is a canonical way of getting a factorisation system in the category of arrows of any category (cf. Korostenski-Tholen), there is a canonical way of getting an ambifibration over a factorisation system $(A,B)$, provided these conditions hold: the pullback of $B$-arrows along $A$-arrows are again in $B$, and the pushout of $A$-arrows along $B$-arrows are again in $A$. Under these conditions, the category of arrows is an ambifibration. (The fat Delta is a variation of this construction: it can be described as the subcategory of the category of arrows of $\Delta$ whose objects are the epis in $\Delta$, and whose arrows are the monos in $Arr(\Delta)$.)

Re synthetic factorisation systems

As Tom mentions, strict factorisation systems is the same thing as distributive laws on monads in $Span$. Here are some further remarks about that, also dating from discussions with Joyal in 2003 (we were unaware of the Rosebrugh-Wood paper).

Strict factorisation systems (on categories with object set $S$) are the same thing as distributive laws between monads in $Span$ based on the set $S$ (which we called commutation laws on monoids in the category of discrete endo-distributors over $S$).

(With this viewpoint one can capture many examples that don’t a priori look like factorisation systems: for example, if given an $S$-category and two subcategories $A$ and $B$ such that pullbacks of $B$-arrows along $A$-arrows exists and are again in $B$, then this amounts to having a commutation law between $A$ and $B^{op}$.)

A strict factorisation system can always be saturated to a ortogonal factorisation system, simply by adding all the isomorphisms to each class. However, strict factorisation systems sometimes contain much more information. For example, the semidirect product of two groups is a nontrivial strict factorisation system, but its saturation is just the chaotic factorisation system!

Furthermore, there are intermediate notions, where the groupoid $A \cap B$ lies in between id and iso. For example, Grothendieck fibrations are such: the intersection is the class of vertical isomorphisms.

All this can be captured in the setting of commutation laws. If $(A,B)$ is a non-strict factorisation system, meaning that the intersection $J := A \cap B$ is bigger than $S$ but smaller than the groupoid of all isos, then $A$ and $B$ become $J$-algebras (in the sense of algebra, i.e. they receive a monoid map from $J$), and then we can consider the tensor product of $J$-algebras: it is just the coequaliser of the two maps $$A \otimes J \otimes B \stackrel{\to}{\to} A \otimes B,$$ just like in algebra. The structure of a factorisation system amounts to a commutation law $B \otimes_J A \to A \otimes_J B$, and hence a monoid structure on the latter object, i.e. a category. Provided each $J \to A$ and $J \to B$ are flat, this category will be the original one. (Flatness is implied if just $J \to A$ and $J \to B$ are monomorphisms of monoids.) If $J$ is the groupoid of all isos, then the tensor product says precisely that we can move isos from the left-hand factor over to the right-hand factor, just as we are used to do in a orthogonal factorisation system.

A ternary factorisation is indeed the same thing as three categories with the same set of objects, two commutation laws satisfying the Yang-Baxter identity — provided the pairwise intersections coincide at some groupoid $J$, and the three categories are flat over $J$.

Coming back to binary factorisation systems: without the flatness condition, as you can guess, some sort of derived tensor product should be used instead, requiring some higher-categorical setting. An important example of non-flatness is provided by the category of spans. Every span factors as one where the second leg is invertible followed by one where the first leg is invertible. The intersection is the class $W$ of all isomorphisms, but the two classes of functors are not flat over $W$, as $W$ acts non-freely on the first class on the right, as well as non-freely on second class on the left.

This fact lies at the heart of the coherence problems for symmetric monoidal categories. Namely, certain spans do admit unique factorisation, and they are precisely the relations. (But the span composite of two relations is not in general a relation.) When selecting coherence diagrams for symmetric monoidal categories, not all diagrams that could commute just for their shape should actually be required to commute. Laplaza first identified those that should (or those that should not), and it was later clarified by Fiore, Hu, and Kriz. The diagrams required to commute are those that correspond to relations, while those corresponding to non-relation spans, i.e. those for which the factorisation is not unique, cannot be required to commute. In other words, the non-flatness accounts for the coherence problems.

(I read what I just wrote, and realise it is not very clear. To see the relationship, one needs first to notice that the category of finite sets and isomorphism classes of finite spans is the Lawvere theory for commutative monoids. To categorify this results, one runs into the problem that some objects, the spans that are not relations, have non-trivial automorphisms (or equivalently, non-unique factorisations). The easiest example is the span $$1 \leftarrow 2 \rightarrow 1,$$ with the nontrivial involution of $2$. This corresponds precisely to the first counter-example to naive coherence for symmetric monoidal categories, namely $a \otimes a$, which has the involution different from the identity. Only rigid spans (i.e. relations) should be used when ‘selecting Laplaza sets’ in the terminology of Fiore-Hu-Kriz.

The span explanation is due to Brun, Carlson, and Dundas, but I don’t think it has actually been proved that this explanation agrees with what Fiore-Hu-Kriz do, so take all this with a grain of salt!

If one just kills the auts, one gets only the degenerate symmetric monoidal categories (i.e. those equivalent to commutative monoids in Cat), as shown by Gould.

The result that spans form the Lawvere theory for commutative monoids has an interesting generalisation, attributed to Tambara (cf. [Gambino-Kock]). Namely, the category of finite sets and isomorphism classes of finite polynomial functors is the Lawvere theory for commutative semirings.

So now I succeeded in digressing into polynomial functors again, so I guess it is time to stop.

Cheers, Joachim.

Thirteen years later, I have finally realized that strict ternary factorization systems are not equivalent to triple distributive laws in $Span$. The problem is that in general, $R_2 L_1$ may not be closed under composition.

In a triple distributive law there are three monads $A,B,C$ and distributive laws $B A \to A B$, $C A \to A C$, and $C B\to B C$ satisfying the Yang-Baxter equation. Thus, in particular there are three composite monads $A B$, $A C$, and $B C$, and the Yang-Baxter equation ensures we get distributive laws $(B C)A\to A(B C)$ and $C(A B) \to (A B)C$ and a unique induced monad structure on $A B C$.

In a ternary factorization system, we would naturally take $A = R_2$, $B = L_2 \cap R_1$, and $C = L_1$. We then get distributive laws $B A \to A B$ and $C B \to B C$, since $A B = R_1$ and $B C = L_2$ are closed under composition. However, if $A C = R_2L_1$ is not closed under composition, we do not get a distributive law $C A \to A C$.

A concrete example of a ternary factorization system where $R_2 L_1$ is not closed under composition is $Top$, with $L_1 =$ quotient maps, $L_2 =$ surjections, $R_1 =$ injections, and $R_2 =$ subspace inclusions. Since any continuous map $X\to Y$ factors as a subspace inclusion $X\to X+Y$ followed by a quotient map $X+Y \to Y$, if $R_2 L_1$ were closed under composition then it would contain every map, which is not the case since there are noninvertible continuous bijections. (This argument is from the Poltr-Tholen paper.)

I expect this is the only obstruction, however: strict ternary factorization systems where $R_2 L_1$ is closed under composition should be equivalent to triple distributive laws.