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.

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