Internal bicategories in additive categories with kernels
by
N. Martins-Ferreira
Polytechnic Institute of Leiria, Portugal

Let A be an additive category with kernels. We construct an equivalence between the category Bicat(A) of internal bicategories in A and the category of all diagrams in A of the form:

Z-s --> Y-t --> X,     l, r : Y --> Z, h : X --> Z

with ts = 0.

In particular every internal bicategory B in the category of abelian groups is obtained (up to an isomorphism) from a diagram of the form above as follows:

• the group of objects in B is X;
• the group of 1-cells is Y \oplusX (the direct sum of Y and X), and (y, x) : x --> t(y)+x in B;
• the composite of (y, x) : x --> t(y)+x with (y', t(y)+x) : t(y)+x --> t(y')+t(y)+x is (y'-sr(y')+y-sl(y)-sh(x), x);
• the identity 1-cell of x is (0, x);
• the group of 2-cells is Z \oplusY \oplusX, and (z, y, x) : (y, x) --> (s(z)+y, x) in B;
• the vertical composite (z, y, x) : (y, x) --> (s(z)+y, x) with (z', s(z)+y, x) : (s(z)+y, x) --> (s(z')+s(z)+y, x) is: (z'+z, y, x);
• hence the identity 2-cell of (y, x) for the vertical composite is (0, y, x);
• the horizontal composite of (z, y, x) : (y, x) --> (s(z)+y, x) with (z', y', t(y)+x) : (y', t(y)+x) --> (s(z')+y', t(y)+x) is: (z'-rs(z')+z-ls(z), y'-sr(y')+y-sl(y)-sh(x), x);
• the associativity isomorphism (y'', t(y')+t(y)+x)((y', t(y)+x)(y, x)) --> ((y'', t(y')+t(y)+x)(y', t(y)+x))(y, x) is: (rsr(y'')-r(y'')+rsl(y')-lsr(y')+l(y)-lsl(y)-ht(y)+rsht(y)+rsh(x)-lsh(x), y''-sr(y'')+y'-sl(y')-sr(y')+slsr(y')+y-2sl(y)+slsl(y)-2sh(x)+slsh(x), x);
• the left identity isomorphism (0, t(y)+x)(y, x) --> (y, x) is (l(y)+h(x), y-sl(y)-sh(x), x);
• the right identity isomorphism (y, x)(0, x) --> (y, x) is (r(y)+h(x), y-sr(y)-sh(x), x).

Date received: May 31, 2002