For the 2-monad $((-)^2,I,C)$ on CAT, with unit $I$ described by identities and multiplication $C$ described by composition, we show that a functor $F : {\cal K}^2 \rightarrow \cal K$ satisfying $FI_{\cal K} = 1_{\cal K}$ admits a unique, normal, pseudo-algebra structure for $(-)^2$ if and only if there is a mere natural isomorphism $F F^2 \rightarrow F C_{\cal K}$. We show that when this is the case the set of all natural transformations $F F^2 \rightarrow F C_{\cal K}$ forms a commutative monoid isomorphic to the centre of $\cal K$.
Theory and Applications of Categories, Vol. 10, 2002, No. 6, pp 134-147 http://www.tac.mta.ca/tac/volumes/10/6/10-06.dvi http://www.tac.mta.ca/tac/volumes/10/6/10-06.ps http://www.tac.mta.ca/tac/volumes/10/6/10-06.pdf ftp://ftp.tac.mta.ca/pub/tac/html/volumes/10/6/10-06.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/10/6/10-06.psTAC Home