# What do you call a map of spaces which is weakly left orthogonal to all $n$-connected maps?

$$\let\op=\operatorname$$In $$\op{Set}$$, we have an $$(\op{Epi},\op{Mono})$$ orthogonal factorization system. Strikingly, if we reverse the roles, we get the no-less-important $$(\op{Mono},\op{Epi})$$ weak factorization system.

In the $$\infty$$-category $$\op{Spaces}$$ of spaces, the most direct analog of the $$(\op{Epi},\op{Mono})$$ orthogonal factorization system on $$\op{Set}$$ is the $$(\text{Effective Epi}, \op{Mono})$$ orthogonal factorization system, but this is just the $$(-1)$$th in a whole tower: for each $$n \in \mathbb Z_{\geq -2}$$, we have an $$(\text{n-connected}, \text{n-truncated})$$ factorization system [1].

It seems that, just as in the analogous case in $$\op{Set}$$, one can take the left half of each these orthogonal factorization systems, and view it as the right half of a weak factorization system $$(\mathcal L_n, \text{n-connected})$$ [2]. To see this, one shows that the $$n$$-connected maps are precisely the maps which are weakly right orthogonal to the maps $$\{S^k \to 1 \mid -1 \leq k \leq n\}$$, and applies the small object argument to obtain factorizations.

In $$\op{Set}$$, we have the cute fact that the resulting weak factorization system $$(\op{Mono},\op{Epi})$$ is just the original orthogonal factorization system $$(\op{Epi},\op{Mono})$$ with the left and right classes swapped. This is not the case in $$\op{Spaces}$$, even when $$n=-1$$: a map $$A \xrightarrow i B$$ of spaces is weakly left orthogonal to the effective epimorphisms if and only if it is a coproduct inclusion $$A \to A \amalg S$$ where $$S$$ is discrete; this is more restrictive than being a monomorphism [3]. I don't know how to characterize the left class $$\mathcal L_n$$ for $$n\geq 0$$ as cleanly. In fact, unlike the case in $$\op{Set}$$, I don't think we have either containment $$\mathcal L_n \subseteq \text{n-truncated}$$ or $$\text{n-truncated} \subseteq \mathcal L_n$$ in general. This leads to my

Questions: Let $$n \in \mathbb Z_{\geq -2}$$.

1. Is there a good characterization of the class of maps $$\mathcal L_n$$, i.e., the maps of spaces which are weakly left orthogonal to the $$n$$-connected maps?

2. What would be a good name for the maps of $$\mathcal L_n$$?

Note that by the small object argument, the maps of $$\mathcal L_n$$ are precisely the retracts of transfinite composites of cobase-changes of coproducts of the maps $$\{S^k \to 1 \mid -1 \leq k \leq n\}$$. So in some sense this is a quite explicit class of maps. By "characterization" I suppose I mean something which can be "checked directly" without having to find all the data of a construction of this form.

[1] Here a map is said to be $$n$$-truncated or $$n$$-connected if its fibers are all so. This convention is off by one from the most classical convention.

[2] Some care should be taken with the definition of a weak factorization system $$\infty$$-categorically: say that a morphism $$A \xrightarrow i B$$ is weakly orthogonal to a morphism $$X \xrightarrow p Y$$ if the map $$\op{Hom}(B,X) \to \op{Hom}(B,Y) \times_{\op{Hom}(A,Y)} \op{Hom}(A,X)$$ is an effective epimorphism. Spelled out, this says that if we have a commutative square—i.e., morphisms $$A \xrightarrow u X$$, $$B \xrightarrow v Y$$ along with a homotopy $$\gamma: pu \sim vi$$, then there exists a lift, i.e., $$B \xrightarrow w X$$ and homotopies $$\alpha: wi \sim u$$, $$\beta: pw \sim v$$ and (here's the only subtle part) a homotopy of homotopies from the composite $$\beta \ast \alpha$$ to $$\gamma$$. Then a weak factorization system is, as usual, a pair of classes of morphisms $$(\mathcal L, \mathcal R)$$ which are complements to each other with respect to weak orthogonality, such that every morphism admits a factorization as a morphism in $$\mathcal L$$ followed by a morphism in $$\mathcal R$$.

[3] Recall that a monomorphism of spaces is a coproduct inclusion $$A \to A \amalg S$$ where $$S$$ may be an arbitrary space.

• Good grief. $\mathcal L_n$ is just the class of maps which are retracts of relative cell complexes of dimension $\leq n+1$. I'd probably call these "retracts of $n+1$-skeletal maps" or something. – Tim Campion Feb 15 '20 at 18:25
• Is this your answer to your own question? I have trouble parsing the meaning of "good grief" here …. – LSpice Feb 15 '20 at 22:17
• @LSpice. Yeah, I think I answered my own question... Maybe I should delete it, but I spent so much time typing it out... And now you've invested time in prettying it up, too! (Thanks, btw) Actually, I suppose it's still an interesting question how to characterize these maps, but $\mathcal L_n$ is so familiar that if a good characterization were known it would have been known in the '50's, and I don't think such a thing is known... Although maybe in the simply-connected case you can say something about dimension vs. homology dimension or something... – Tim Campion Feb 15 '20 at 22:18
• To clarify, "Good grief" is my sheepish exclamation when I realized that I found a ridiculous roundabout way to ask for a characterization of the retracts of relative $n+1$-dimensional complexes without recognizing that that's what I was asking! – Tim Campion Feb 15 '20 at 22:28
• Yes, there is a characterization in terms of (co)homology in most cases. Of course for a map in this class the relative homology and cohomology vanishes in degrees above $n+1$. Conversely, if relative $H^j$ vanishes for all $j\ge n+1$, for all coefficient systems on the codomain, then the map is in that class. This holds for all $n>2$, at least. – Tom Goodwillie Feb 15 '20 at 23:46

This question was answered in the comments $$\mathcal L_n$$ comprises those maps which are retracts of relative $$\leq n+1$$-dimensional relative cell complexes.
Tom Goodwillie explains in the comments a cohomological characterization of $$\mathcal L_n$$ for sufficiently large $$n$$.