A384135 -id:A384135 - cp's OEIS frontend

A384134 Triangle read by rows: T(n,k) is the number of Cauchy-complete categories with n morphisms and k objects.

1, 1, 1, 1, 2, 1, 2, 6, 2, 1, 1, 12, 9, 2, 1, 2, 23, 25, 10, 2, 1, 1, 45, 69, 35, 10, 2, 1, 5, 98, 178, 119, 38, 10, 2, 1, 2, 278, 457, 371, 151, 39, 10, 2, 1

Offset: 1

Elijah Beregovsky, May 20 2025

A Cauchy complete (also called Karoubi complete or idempotent-complete) category is one in which all idempotents split. In other words, in a Cauchy-complete category every arrow e:A->A such that e=e*e has a retract, meaning there exists an object B and morphisms r:A→B and s:B→A such that s∘r=e but r∘s=1_B.

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  2,  6,  2,  1;
  1, 12,  9,  2, 1;
  2, 23, 25, 10, 2, 1;
  ...

Geoff Cruttwell, Counting Finite Categories, presentation, (2018).
nLab, Cauchy complete category.
nLab, Karoubi envelope.

Cf. A384135 (row sums), A000001 (column 1), A384066 (limiting values), A125697.

T(n,k) = A384066(n-k) if k >= (2/3)*n.
T(3n,2n) = T(3n-1,2n-1) + 1 when n >= 1.
T(3n-1,2n-1) = T(3n-2,2n-2) + 3 when n >= 2.
T(3n-2,2n-2) = T(3n-3,2n-3) + 13 when n >= 4.

cp's OEIS Frontend

A384134 Triangle read by rows: T(n,k) is the number of Cauchy-complete categories with n morphisms and k objects.

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