It is easy to see that factorizations and partitions are duals if we model them in the category FinSet.
A partition on a set S is “just” an epimorphism (i.e., a surjection in FinSet), e:S↠X. That’s because an epimorphism induces a partition on S=∐x∈Xe−1(x) indexed by the elements x∈X. (Surjectivity/epicness is necessary because without it we would have some x’s beyond the image of e, so that their preimages would be empty: e−1(x)=∅). In the other direction, any partition S=∐i∈IXi induces a unique surjection mapping each element of S to the part it belongs to. It’s easy to see that these two views are equivalent (i.e. moving partition→epimorphism→partition gets us back to the same partition and similarly epimorphism→partition→epimorphism).
So, a factorization of a set S is an isomorphism e:S∼→∏i∈Ibi constructed as a product of surjections indexed by a (finite) set I, e=∏i∈Iei, ∀i∈I.ei:S↠bi. Explicitly: e(s)=⟨e1(s),e2(s),...,e|I|(s)⟩. Moreover, we require each partition/epic ei:S↠bi to be non-trivial, i.e., it must have at least two elements, so none of the bi’s is a singleton, i.e., the terminal object.
If we dualize this construction, we get an isomorphism e:∐i∈Ibi∼→S from the coproduct (i.e., disjoint sum in FinSet) to the set S, that is constructed from monomorphisms (i.e., injections in FinSet) e=∐i∈Iei, ∀i∈I.ei:bi↣S. Moreover, since in the factorization case we assumed that none of the bi‘s is terminal (singleton), here, after dualization, none of the bi’s is initial, i.e., it is not the empty set. The isomorphism means that the two sets are equinumerous: |∐i∈Ibi|=∑i∈I|bi|=|S|, so the set of the “co-basis” elements (bi)i∈I is isomorphic to a partition of S, since, as we just remarked, each bi is non-empty. In other words, the “co-basis” elements “are” parts of a partition. Each ei being an injection means that ∀i∈I.|bi|≤|S|, but that’s already implied by equinumerosity (actually, strict inequality is implied because each bi is non-empty). The natural interpretation of the monic ei is the subset inclusion of the elements of the part bi⊂S.
To go from the partition-as-epi view e:S↠I, we “convert” it into the isomorphism between S and the disjoint union of the parts of the partition ∐i∈Ibi∼→S (where bi=e−1(i)), which can be viewed as a coproduct of subset inclusions (i.e. monics/injections), and then dualize to get S∼→∏i∈Ibi.
[Previously in categorical view of FFS: drocta and Gurkenglas. Most likely somebody has figured this out already, but I haven’t seen it written up anywhere, so I’m posting this comment.]
It is easy to see that factorizations and partitions are duals if we model them in the category FinSet.
A partition on a set S is “just” an epimorphism (i.e., a surjection in FinSet), e:S↠X. That’s because an epimorphism induces a partition on S=∐x∈Xe−1(x) indexed by the elements x∈X. (Surjectivity/epicness is necessary because without it we would have some x’s beyond the image of e, so that their preimages would be empty: e−1(x)=∅). In the other direction, any partition S=∐i∈IXi induces a unique surjection mapping each element of S to the part it belongs to. It’s easy to see that these two views are equivalent (i.e. moving partition→epimorphism→partition gets us back to the same partition and similarly epimorphism→partition→epimorphism).
So, a factorization of a set S is an isomorphism e:S∼→∏i∈Ibi constructed as a product of surjections indexed by a (finite) set I, e=∏i∈Iei, ∀i∈I.ei:S↠bi. Explicitly: e(s)=⟨e1(s),e2(s),...,e|I|(s)⟩. Moreover, we require each partition/epic ei:S↠bi to be non-trivial, i.e., it must have at least two elements, so none of the bi’s is a singleton, i.e., the terminal object.
If we dualize this construction, we get an isomorphism e:∐i∈Ibi∼→S from the coproduct (i.e., disjoint sum in FinSet) to the set S, that is constructed from monomorphisms (i.e., injections in FinSet) e=∐i∈Iei, ∀i∈I.ei:bi↣S. Moreover, since in the factorization case we assumed that none of the bi‘s is terminal (singleton), here, after dualization, none of the bi’s is initial, i.e., it is not the empty set. The isomorphism means that the two sets are equinumerous: |∐i∈Ibi|=∑i∈I|bi|=|S|, so the set of the “co-basis” elements (bi)i∈I is isomorphic to a partition of S, since, as we just remarked, each bi is non-empty. In other words, the “co-basis” elements “are” parts of a partition. Each ei being an injection means that ∀i∈I.|bi|≤|S|, but that’s already implied by equinumerosity (actually, strict inequality is implied because each bi is non-empty). The natural interpretation of the monic ei is the subset inclusion of the elements of the part bi⊂S.
To go from the partition-as-epi view e:S↠I, we “convert” it into the isomorphism between S and the disjoint union of the parts of the partition ∐i∈Ibi∼→S (where bi=e−1(i)), which can be viewed as a coproduct of subset inclusions (i.e. monics/injections), and then dualize to get S∼→∏i∈Ibi.
[Previously in categorical view of FFS: drocta and Gurkenglas. Most likely somebody has figured this out already, but I haven’t seen it written up anywhere, so I’m posting this comment.]