Page 118 - Textos de Matemática Vol. 45

110 Chapter 9. Appendix 9.7 The shift functor
Any category X1 produces a category Dec1X1, given by the upper row along with a downward vertical functor, denoted by ε1X1 (see [7]):
d0
X4
X3
oo X3
d0
s0 d0
//
d1 // X2 oo s0 X1
LL oo LL
s1 d1
//
d4 d3 d2 d1
d3 d0
d2 d0
oo X2
X
// X
s0 d0
d1 // 1oos 0
0 s1 d1
LL oo LL d2
//
d3
where X4 is given by the following pullback, which defines X4 as the “objects
of quadruple composables pairs”.
X d0 // X (9.7.3)
43
X1 is a groupoid if and only if so is Dec1X1.
d4 X3
d3 X2,
// d0