Page 25 - Textos de Matemática Vol. 46

COLIMITS OF MONADS 15
This is well-defined because eA ⇤ eA = eRA · ReA is a epimorphism. To verify the unit axioms μR · ⌘R = id, consider the following diagram:
SA eA // RA mA // TA
⌘S ⌘R ⌘T SA RA TA
μSA
✏✏
// ✏✏ mA⇤mA μRA
axiom. Naturality of ⌘S implies that the upper left-hand square commutes: eRA ·SeA ·⌘SA =eRA ·⌘RSA ·eA =⌘AR ·eA.
Analogously for the upper right-hand square. Consequently, the diagonal pas- sagefromSAtoTAintheabovediagramsatisfies(duetoμTA·⌘TA =id)the equality
mA ·(μRA ·⌘RA)·eA =mA ·eA.
Since nA is strongly monic and eA epic, this implies μRA · ⌘RA = id .
The verification of the other unit axiom μR · R⌘R = id is analogous. The proof of the associativity
μR · RμR = μR · μRR follows from the following diagram:
SSSA eA⇤eA⇤eA // RRRA mA⇤mA⇤mA // TTTA μ S S A μ SA μ R R A R μ RA μ T T A T μ TA
✏✏ eA⇤eA
SSA RRA TTA
// ✏✏ // ✏✏
// ✏✏
RA mA
μTA TA
SA eA
Its outward square commutes since S and T both satisfy the corresponding
✏✏ ✏✏ eA⇤eA ✏✏
// ✏✏ ✏✏ mA⇤mA μRA
// ✏✏
RA mA
// ✏✏ ✏✏ SSA RRA TTA
μSA
SA eA
μTA TA
// ✏✏