Orthogonal morphisms

Question

In category theory, there's a notion of orthogonality between two (not-necessarily parallel) morphisms. It seems to pretty important due to its connection to factorization systems, but I don't really get why this notion is natural, what it "means" in the deeper, philosophical sense, nor how to visualize or otherwise understand it.

Question. To those who feel they understand orthogonality: where did you learn about it, what do you think/visualize/see inside your brain when you hear this word, and why is it natural to consider this notion in the context of factorization systems?

Answer 1

I feel I have to take the challenge.

It is difficult to date precisely when the notion of orthogonality arose in category theory; an embryo of the (Epi, Mono)-factorization system on $\bf Grp$ was outlined in Mac Lane's "Groups, Categories and Duality", dated 1948 (!!!), but it was only Isbell (if I remember well; he called a factorization system a "bicategorical structure": easy to see why this nomenclature was abandoned) that noticed the classes of maps of a factorization system were "orthogonal" in modern sense.

My feeling is that for one reason or the other factorization systems have always been around, since the very beginning of CT. It is then natural

1. that most authors kept "inventing the wheel" independently finding since the very beginning various properties lacking a general, unitary foundation; as a consequence, you find several slightly different approaches to the topic that make you feel you're missing something.
2. that orthogonality/factorization are ubiquitous properties in category-like structure. As a consequence, there are well-established and flokloristic reasons why you should be interested in these properties when studying a category, but the motivations are kept under the carpet. Nowadays everybody works with simplicial sets, but how many people in the new generation can reconstruct the "archaic" geometric/combinatorical intuition that strongly motivates them?

Anyways. This is an answer to your question, not a tirade against the new generation. I'm part of it, maybe ☺

The orthogonality relation, linked or not to FS theory, is so important because

1. It is ubiquitous: several "things" can be described via, or involve orthogonality relation.
2. It can be generalized at low cost to different approaches at CT: suppose you like good old "working math" category theory. Suppose you like higher category theory. Suppose you like enriched category theory... you can everywhere define the orthogonality relation between "arrows" (whatever they mean) and employ it to retrieve a "calculus" of these things.

As examples of 1:

• Ok, there's this relation on a category; it is well-known that every relation induces a Galois connection between powersets: here you send $\mathcal H$ into $\mathcal H^\perp = \{g\mid h\perp g\forall h\in \mathcal H\}$, or ${}^\perp\mathcal H$ (dually defined). Fine. Now, nearly every class which is worth studying in usual category theory can be defined as $\mathcal H^\perp$ or ${}^\perp\mathcal H$ for some $\cal H$. Monomorphisms? Check. Epimorphisms? Check. Isomorphisms? Check. Projective morphisms of $R$-modules? Check. Serre fibration of spaces? Check. Functors injective on objects? Check. Also (it's not unrelated, obviously), several properties of objects $X$ can be expressed as "the initial/terminal morphism of $X$ lies in $\mathcal H^\perp$ or ${}^\perp\mathcal H$ for some $\cal H$. Being projective/injective? Check. Being a Cauchy complete category? Check. Being a groupoid? Check.
• A quite general problem in category theory called the "orthogonal subcategory problem" can be adressed within the framework of factorization systems on suitably nice categories: it is explained here.
• In these days, it's impossible not to stumble into model categories if you do a certain kind of Mathematics. It is said in Whitehead's "Elements of homotopy theory" that algebraic topology ultimately deals with problems of extension and lifting of maps. But these are precisely the kind of problems that orthogonality adresses by design! It appears extremely natural that everywhere you can state a problem in algebraic topology ispired by, or leading to, an extension/lifting problem, there you can use the power of orthogonality relation. Now, name a single problem in algebraic topology that doesn't seem an extension/lifting problem in disguise (just kidding; there are many, but there's many ext/lift problems as well).
• Under really mild assumptions, any reflective subcategory $\mathcal A \subseteq \mathcal C$ arises from a FS on $\cal C$, and there's an explicit recipe to turn this into a bijection, in some cases. see here

As examples of 2:

• Suppose you live in an enriched category, say a $\cal V$-category (topological spaces, chain complexes, simplicial sets, $[0,\infty]$ are perfectly fine examples of $\cal V$). Then you can say that two morphisms are orthogonal if the canonical map $$ \hom(B,X) \xrightarrow{\qquad} \hom(A,X)\times_{\hom(A,Y)}\hom(B,Y) $$ is an isomorphism in $\cal V$ (easy exercise: if $\cal V=\bf Set$ then this is equivalent to the "fill the square" property).
• If you live in a quasicategory, read chapter 1 of this paper.

(If time permits, at some point I'll add references and hyperlinks for other things I said.)

Answer 2

My intuition about a (weakly, homotopically, ...) orthogonal pair $f:A\to B,g:X\to Y$ tends to assume either $Y=*$ is terminal or $A=\emptyset$ is initial. Then, piggybacking off the end of the previous answer, saying $f$ and $g$ are orthogonal is just saying the composition morphism $\mathrm{Hom}(B,X)\to\mathrm{Hom}(A,X)$, respectively $\mathrm{Hom}(B,X)\to\mathrm{Hom}(B,Y)$, is an isomorphism (epimorphism, weak equivalence, ...) In a sense, $f$ is left orthogonal to $X\to *$ if $X$ thinks $f$ is an isomorphism (etc.) In homotopy theory, we say $X$ is an $f$-local object, and by judicious choice of $f$ we might have found for $X$, for instance, a space whose homotopy groups are rational vector spaces ($X$ thinks rational homotopy equivalences are homotopy equivalences.) Then orthogonality in squares becomes just a relative form of the orthogonality above, which is essentially all in triangles. This also looks more like the early motivational examples of injective and projective objects.