  In this paper, we show that for reduced homotopy endofunctors of spaces, F,
and for all $n \geq 1$ there are adjoint functors $R_n, L_n$ with $T_n F \simeq
R_n F L_n$, where $P_n F$ is the $n$-excisive approximation to $F$, constructed
by taking the homotopy colimit over iterations of $T_n F$. This then endows
$T_n$ of the identity with the structure of a monad and the $T_n F$'s are the
functor version of bimodules over that monad. It follows that each $T_n F$ (and
$P_nF$) takes values in spaces of symmetric Lusternik-Schnirelman cocategory
$n$, as defined by Hopkins. This also recovers recent results of
Chorny-Scherer. The spaces $T_n F(X)$ are in fact classically nilpotent (in the
sense of Berstein-Ganea) but not nilpotent in the sense of Biedermann and
Dwyer. We extend the original constructions of dual calculus to our setting,
establishing the $n$-co-excisive approximation for a functor, and dualize our
constructions to obtain analogous results concerning constructions $T^n$,
$P^n$,and LS category.
