#
The waves of a total category

##
R.J. Wood

For any *total* category $K$, with defining adjunction $\sup \ladj Y
: K \rightarrow set^{K^{op}}$, the expression $W(a)(k)=
set^{set^{K^{op}}}(K(a,\sup -),[k,-])$, where $[k,-]$ is evaluation at
$k$, provides a well-defined functor $W : K \rightarrow \hat{K} =
set^{K^{op}}$. Also, there are natural transformations $\beta : W\sup
\rightarrow 1_{\hat{K}}$ and $\gamma : \sup W \rightarrow 1_K$ satisfying
$\sup\beta =\gamma\sup$ and $\beta W =W\gamma$. A total $K$ is *totally
distributive* if $\sup$ has a left adjoint. We show that $K$ is
totally distributive iff $\gamma$ is invertible iff $W \ladj \sup$. The
elements of $W(a)(k)$ are called *waves from $k$ to $a$.*

Write $\tilde{K}(k,a)$ for $W(a)(k)$. For any total $K$ there is an
associative composition of waves. Composition becomes an arrow $\bullet :
\tilde{K}\circ_{K}\tilde{K} \rightarrow \tilde{K}$. Also, there is an
augmentation $\tilde{K}(-,-) \rightarrow K(-,-)$ corresponding to a
natural $\delta : W \rightarrow Y$ constructed via $\beta$. We show that
if $K$ is totally distributive then $\bullet$ is invertible and then
$\tilde{K}$ supports an idempotent comonad structure. In fact, $\tilde{K}
\circ_{K} \tilde{K} = \tilde{K} \circ_{\tilde{K}} \tilde{K}$ so that
$\bullet$ is the coequalizer of $\bullet K$ and $K \bullet$, making
$\tilde{K}$ a *taxon* in the sense of Koslowski. For a small taxon
$T$, the category of interpolative modules $iMod(1,T)$ is totally
distributive. Here we show, for any totally distributive $K$, that there
is an equivalence $K \rightarrow iMod(1,\tilde{K})$.

Keywords:
adjunction, totally cocomplete, totally distributive,
taxon, i-module, proarrow equipment

2010 MSC:
18A30

*Theory and Applications of Categories,*
Vol. 30, 2015,
No. 47, pp 1624-1646.

Published 2015-11-30.

http://www.tac.mta.ca/tac/volumes/30/47/30-47.pdf

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