# Intuition behind orthogonality in category theory, and origin of name

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.

• According to this page, "orthogonal factorization systems" used to be called simply "factorization systems". – Najib Idrissi Feb 6 '19 at 13:19

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.)