Original: Theorem H.11.
Revised: Theorem H.11.

Original: Let α : X → W be a morphism in I ( Q L ) .
Revised: Let α : X → W be a morphism in I ( Q L ) .

Original: Let Q be an orthogonal factorization of α and set U
Revised: Let Q be an orthogonal factorization of α and set U

Original: = W − Q · X .
Revised: = W − Q · X .

Original: The representation Q − 1 · W = X + Q − 1 · U belongs to Rep int ( m , n ) .
Revised: The representation Q − 1 · W = X + Q − 1 · U belongs to Rep int ( m , n ) .

Original: For any k ≥ 0 , we have
Revised: For any k ≥ 0 , we have

Original: [Equation]
Revised: [Equation]

Original: For any k ≥ 0 , we have
Revised: For any k ≥ 0 , we have

Original: [Equation]
Revised: [Equation]

Original: H. R ELATION TO A LGORITHM
Revised: H. R ELATION TO A LGORITHM

Original: In this ﬁnal section, we relate the general categorical results of this appendix to Algorithm 1.
Revised: In this ﬁnal section, we relate the general categorical results of this appendix to Algorithm 1.

Original: Recall that, in Section 5.2, we associated a reduced dimension vector n red to any dimension vector n of the neural quiver Q L .
Revised: Recall that, in Section 4.2, we associated a reduced dimension vector n red to any dimension vector n of the neural quiver Q L .

Original: Proposition H.12.
Revised: Proposition H.12.

Original: Let W be a representation of Q L with dimension vector n .
Revised: Let W be a representation of Q L with dimension vector n .

Original: Then W admits a morphism in I ( Q L ) from a representation with dimension vector n red .
Revised: Then W admits a morphism in I ( Q L ) from a representation with dimension vector n red .

Original: Proof.
Revised: Proof.

Original: We proceed by induction on the number N of i ∈ { 1 , . . .
Revised: We proceed by induction on the number N of i ∈ { 1 , . . .

Original: , L − 1 } such that n red i < n i .
Revised: , L − 1 } such that n red i < n i .

Original: If
Revised: If

Original: N = 0 , there is nothing to show.
Revised: N = 0 , there is nothing to show.

Original: Otherwise, let j be the smallest element of { 1 , . . .
Revised: Otherwise, let j be the smallest element of { 1 , . . .

Original: , L − 1 } such that n red j < n j .
Revised: , L − 1 } such that n red j < n j .

Original: Then n red j = n j − 1 + 1 .
Revised: Then n red j = n j − 1 + 1 .

Original: Let W j = Q ◦ inc 1+ n j − 1 ,n j ◦ R be a QR decomposition of W , where Q ∈ O ( n j ) and R is an upper-triangular n j − 1 + 1 by n j − 1 + 1 matrix.
Revised: Let W j = Q ◦ inc 1+ n j − 1 ,n j ◦ R be a QR decomposition of W , where Q ∈ O ( n j ) and R is an upper-triangular n j − 1 + 1 by n j − 1 + 1 matrix.

Original: Consider the representation of Q L given by:
Revised: Consider the representation of Q L given by:

Original: [Equation]
Revised: [Equation]

Original: Then α = ( id n 1 , . .
Revised: Then α = ( id n 1 , . .

Original: , id n
Revised: , id n

Original: [Equation]
Revised: [Equation]

Original: Q ◦ inc [Equation] n
Revised: Q ◦ inc 1+ n

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: n
Revised: n

Original: [Equation]
Revised: [Equation]

Original: id n
Revised: id n

Original: [Equation]
Revised: [Equation]

Original: +1 , . .
Revised: +1 , . .

Original: , id n L ) is a morphism form X to W .
Revised: , id n L ) is a morphism form X to W .

Original: The dimension vector of X is ( n red0 , . . .
Revised: The dimension vector of X is ( n red0 , . . .

Original: , n red j − 1 , n red j , n j +1 , . .
Revised: , n red j − 1 , n red j , n j +1 , . .

Original: . , n L ) and has one less coordinate that n not equal to n red , so the induction hypothesis applies.
Revised: . , n L ) and has one less coordinate that n not equal to n red , so the induction hypothesis applies.

Original: Consequently, Theorems 5.3 and 5.7 can be viewed as a corollaries of Proposition H.7, Theorem
Revised: Consequently, Theorems 4.3 and 4.7 can be viewed as a corollaries of Proposition H.7, Theorem

Original: H.11, and Proposition H.12.
Revised: H.11, and Proposition H.12.

Original: Remark H.13.
Revised: Remark H.13.

Original: Let Q , R and U be the outputs of Algorithm 1 applied to a representation W of dimension vector n .
Revised: Let Q , R and U be the outputs of Algorithm 1 applied to a representation W of dimension vector n .

Original: Then R deﬁnes a representation of dimension vector n red , and Q deﬁnes a morphism in I ( Q L ) from R to W .
Revised: Then R deﬁnes a representation of dimension vector n red , and Q deﬁnes a morphism in I ( Q L ) from R to W .

Original: Indeed, for i = 1 , . . .
Revised: Indeed, for i = 1 , . . .

Original: , L , Algorithm 1 provides the equality
Revised: , L , Algorithm 1 provides the equality

Original: [Equation]
Revised: [Equation]

Original: i ,
Revised: i ,

Original: (where Q 0 = id n 0 = inc 0 and Q L = id n L = inc L ), which implies that the following diagram commutes:
Revised: (where Q 0 = id n 0 = inc 0 and Q L = id n L = inc L ), which implies that the following diagram commutes:

Original: [Equation]
Revised: [Equation]

Original:  red 
Revised: n red i −

Original: [Equation]

Original: R i
Revised: R i

Original: [Equation]
Revised: [Equation]

Revised: 0 0 Q

Original: [Equation]
Revised: [Equation]

Original: Q
Revised: − 1 (cid:21) ◦

Original: [Equation]
Revised: [Equation]

Original: inc
Revised: inc

Original: [Equation]
Revised: [Equation]

Revised: −

Original: [Equation]
Revised: [Equation]

Revised: (cid:15)

Original: R n red
Revised: R n red

Original: i
Revised: i

Original: Q
Revised: Q

Original: i ◦ inc i [Equation]
Revised: i ◦ inc i (cid:15) (cid:15)

Original: R 1+ n i −
Revised: R 1+ n i −

Original: W i (cid:47) (cid:47) R n i
Revised: W i (cid:47) (cid:47) R n i