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In the following four examples, there is a pattern that tends to have the same template. For objects $A$ and $C$, given any morphism $f:A\to C$ there exists a unique object $B$ with unique morphisms $\varphi:A\to B$ and $\psi:B\to C$ such that $f = \psi\circ\varphi$.

Is there some categorical property/concept connecting these examples? It feels too similar to just be coincidence.

  • For two groups $G,H$ and a homomorphism $f:G\to H$, the unique object is $\ker(f)$.
  • For a group $G$ and an abelian group $H$, the unique object is $[G,G]$.
  • For a commutative monoid $M$ and an abelian group $A$, the unique object is the Grothendieck group.
  • For a topological space $X$ and a compact Hausdorff space $Y$, the unique object is the Stone-Cech compactification $\beta X$.
Santana Afton
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    Have you read the Wikipedia page on "universal properties"? That's exactly what it means. – symplectomorphic Aug 11 '17 at 07:02
  • Oh, yeah that's exactly what I'm looking for. Thanks! Feel free to post that as an answer -- I'd be happy to +1. – Santana Afton Aug 11 '17 at 07:15
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    Usually to ensure uniqueness, the morphisms $\psi,\phi$ are required to have a certain property. For instance the case $G\to G/Ker(f)\to H$ is called an "epi-mono" factorization because the first map is an epimorphism and the second one a monomorphism. In the category of groups, epi-mono factorizations exist and are essentially unique. It's actually the case for all algebras, where $A/Ker(f)$ has a meaning that agrees with tye previous one for groups. In Abstract and concrete Categories, a whole chapter is devoted to the study of so called factorization-structures. – Maxime Ramzi Aug 11 '17 at 07:26
  • Another type of categories in which you have such a factorization is topoi, in which one has (again) an essentially unique epi-mono-factorization. – Maxime Ramzi Aug 11 '17 at 07:27

2 Answers2

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Yes: these are called universal properties. See Wikipedia.

symplectomorphic
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The Stone-Cech compactification functor $\beta$ is the left-adjoint of the inclusion functor CompactHausdorff into Top.

The Grothendieck group functor is left-adjoint to the forgetful functor AbGrp into CommutativeMonoids.

The abelianisation functor is left-adjoint to the the inclusion functor AbGrp into Grp.

To quote Wikipedia since I can't be bothered to type:

Consider the category $\mathcal{D}$ of homomorphisms of abelian groups. If $f_1 : A_1 \to B_1$ and $f_2 : A_2 \to B_2$ are two objects of $\mathcal{D}$, then a morphism from $f_1$ to $f_2$ is a pair $(g_A, g_B)$ of morphisms such that $g_B \circ f_1 = f_2 \circ g_A$. Let $G : \mathcal{D} \to \mathbf{Ab}$ be the functor which assigns to each homomorphism its kernel and let $F : \mathcal{D} ← \mathbf{Ab}$ be the functor which maps the group $A$ to the homomorphism $A → 0$. Then $G$ is right adjoint to $F$.

Patrick Stevens
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