A partition is a family of sets called parts. A partition morphism X->X’ has a function from each part of X to some part of X’. It witnesses that X is finer than X’¹.
The underlying set of a partition is its disjoint union. Call the discrete partition of S DS. The functions S->⊔X correspond to the partition morphisms DS->X. Call the trivial partition of S TS. The functions ⊔X->S correspond to the partition morphisms X->TS. In terser notation, we have D ⊣ ⊔ ⊣ T.
A factorization is a family of partitions called factors. A factorization morphism B->B’ has a partition morphism to each factor of B’from some factor of B.²
The underlying partition of a factorization is its common refinement. Call the trivial factorization of X FX.³ The partition morphisms X->∨B correspond to the factorization morphisms FX->B: We have F ⊣ ∨. The absence of “discrete factorizations” as a right adjoint to ∨ is where histories come from.
A history of ∨B->X is a nice⁴ B->H with a ∨H->X. The history of ∨B->X is its terminal history. Note that this also attempts to coarsen each factor. ∨B->X being weakly after ∨B->X’ is witnessed by a nice ∨H->X’ or equivalently H->H’. ∨B->X and ∨B->X’ are orthogonal iff the pushout of B->H and B->H’ is the empty factorization.
Translating “2b. Conditional Orthogonality” is taking a while (I think it’s something with pushouts) so let’s post this now. I’m also planning to generalize “family” to “diagram”. Everyone’s allowed to ask stupid questions, including basic category theory.
¹: Which includes that X might rule out some worlds. ²: Trying to avert the analogy break cost me ~60% of the time behind this comment. ³: F for free, as in the left adjoint, and as in factorization. ⁴: Nice means that everything in sight commutes.
Let 1 be the category with one object • and one morphism. Let Δx be the constant functor to x.
A set is a family of • called elements. A set morphism S->S’ has a 1-morphism between each element of S and some element of S’. The 1-morphisms •->Δ•S correspond to the set morphisms Δ∅•->S. The 1-morphisms Δ•S->• correspond to the set morphisms S->Δ1•. We have Δ∅ ⊣ Δ• ⊣ Δ1.
Let 0 the empty category. • is the empty family. A 1-morphism has nothing to prove. There’s no forgetful functor 1->0 so the buck stops here.
Call the index set of X IX. Call the partition into empty parts indexed by S 0S. We have 0 ⊣ I ⊣ D ⊣ ⊔ ⊣ T.
None of the our three adjunction strings can be extended further. Let’s apply the construction that gave us histories at the other 5 ends. Niceness is implicit. - The right construction of TS->X is the terminal S->S’ with a TS’->X: The image of ⊔(TS->X). - The left construction of X->0S is the initial S’->S with a X->0S’: The image of I(X->0S). - The left construction of B->FX is the initial X’->X with a B->FX’: The image of ∨(B->FX). - The right construction of Δ1•->S is the terminal •->• with a Δ1•->S: The image of Δ•(Δ1•->S). - The left construction of S->Δ∅• is absurd, but can still be written as the image of Δ•(S->Δ∅•). - The history of ∨B->X is the terminal B->B’ with a ∨B’->X: Breaks the pattern! F(∨B->X) does not have the information to determine the history.
In fact, ⊔T, I0, ∨F, Δ•Δ1 and Δ•Δ∅ are all identity, only F∨ isn’t.
Let’s try category theory.
A partition is a family of sets called parts. A partition morphism X->X’ has a function from each part of X to some part of X’. It witnesses that X is finer than X’¹.
The underlying set of a partition is its disjoint union. Call the discrete partition of S DS. The functions S->⊔X correspond to the partition morphisms DS->X. Call the trivial partition of S TS. The functions ⊔X->S correspond to the partition morphisms X->TS. In terser notation, we have D ⊣ ⊔ ⊣ T.
A factorization is a family of partitions called factors. A factorization morphism B->B’ has a partition morphism to each factor of B’ from some factor of B.²
The underlying partition of a factorization is its common refinement. Call the trivial factorization of X FX.³ The partition morphisms X->∨B correspond to the factorization morphisms FX->B: We have F ⊣ ∨. The absence of “discrete factorizations” as a right adjoint to ∨ is where histories come from.
A history of ∨B->X is a nice⁴ B->H with a ∨H->X. The history of ∨B->X is its terminal history. Note that this also attempts to coarsen each factor. ∨B->X being weakly after ∨B->X’ is witnessed by a nice ∨H->X’ or equivalently H->H’. ∨B->X and ∨B->X’ are orthogonal iff the pushout of B->H and B->H’ is the empty factorization.
Translating “2b. Conditional Orthogonality” is taking a while (I think it’s something with pushouts) so let’s post this now. I’m also planning to generalize “family” to “diagram”. Everyone’s allowed to ask stupid questions, including basic category theory.
¹: Which includes that X might rule out some worlds.
²: Trying to avert the analogy break cost me ~60% of the time behind this comment.
³: F for free, as in the left adjoint, and as in factorization.
⁴: Nice means that everything in sight commutes.
Let 1 be the category with one object • and one morphism. Let Δx be the constant functor to x.
A set is a family of • called elements. A set morphism S->S’ has a 1-morphism between each element of S and some element of S’. The 1-morphisms •->Δ•S correspond to the set morphisms Δ∅•->S. The 1-morphisms Δ•S->• correspond to the set morphisms S->Δ1•. We have Δ∅ ⊣ Δ• ⊣ Δ1.
Let 0 the empty category. • is the empty family. A 1-morphism has nothing to prove. There’s no forgetful functor 1->0 so the buck stops here.
Call the index set of X IX. Call the partition into empty parts indexed by S 0S. We have 0 ⊣ I ⊣ D ⊣ ⊔ ⊣ T.
None of the our three adjunction strings can be extended further. Let’s apply the construction that gave us histories at the other 5 ends. Niceness is implicit.
- The right construction of TS->X is the terminal S->S’ with a TS’->X: The image of ⊔(TS->X).
- The left construction of X->0S is the initial S’->S with a X->0S’: The image of I(X->0S).
- The left construction of B->FX is the initial X’->X with a B->FX’: The image of ∨(B->FX).
- The right construction of Δ1•->S is the terminal •->• with a Δ1•->S: The image of Δ•(Δ1•->S).
- The left construction of S->Δ∅• is absurd, but can still be written as the image of Δ•(S->Δ∅•).
- The history of ∨B->X is the terminal B->B’ with a ∨B’->X: Breaks the pattern! F(∨B->X) does not have the information to determine the history.
In fact, ⊔T, I0, ∨F, Δ•Δ1 and Δ•Δ∅ are all identity, only F∨ isn’t.