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In category theory, two morphisms $e:A\to B$ and $m:C\to D$ are said to be orthogonal if for any $f:A\to C$ and $g:B\to D$ with $m\circ f=g\circ e$, there exists a unique morphism $d:B\to C$ such that $f=d\circ e$ and $g=m\circ d$.

What a possible interpretation of this concept? And why is it called "orthogonal", does this help the intuition in some way? Is there a standard example where the name becomes clear?

An introductory reference would also be welcome.

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  • $\begingroup$ According to this page, "orthogonal factorization systems" used to be called simply "factorization systems". $\endgroup$ – Najib Idrissi Feb 6 '19 at 13:19
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I don't know the origin of the word. (Calling pairs of morphisms, or more generally pairs of classes of morphisms, "orthogonal" certainly long predates the very recent annexation of the adjective onto "orthogonal factorization system" to distinguish it from a "weak factorization system".) However, if I had to guess I would guess that it comes from the following analogy.

  1. Orthogonality between vectors in an inner product space $U$ is a binary relation, which generates a Galois connection on the poset of subsets of the vector space. If $V,W$ are a fixed-pair of the Galois connection (that is, $V^{\perp} = W$ and $W^{\perp}=V$), then for every vector $u$ there is at most one way to write $u = v + w$ with $v\in V$ and $w\in W$. If in addition every vector has such a decomposition, then $U = V\oplus W$ is a direct sum decomposition of the ambient vector space.

  2. Orthogonality between morphisms in a category $C$ is a binary relation, which generates a Galois connection on the poset of subclasses of morphisms of the category. If $E,M$ are a fixed pair of the Galois connection (that is, $E^{\perp} = M$ and $^{\perp}M=E$), then for every morphism $f$ there is at most one way, up to unique isomorphism, to write $f = m \circ e$ with $m\in M$ and $e\in E$. If in addition every morphism has such a decomposition, then $(E,M)$ is an (orthogonal) factorization system. (Indeed, in this case we have "$C = M\circ E$" as monads in $\rm Prof$ via a distributive law, as shown by Cheng --- this long postdates the terminology, but the underlying intuition was probably there already.)

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