nLab totally distributive category

Definition

Recall that a locally small category $\mathcal{K}$ is total if its Yoneda embedding $Y\colon\mathcal{K}\longrightarrow [\mathcal{K}^{op},Set]$ has a left adjoint $X$.

Definition

A total category $\mathcal{K}$ is totally distributive if $X \colon [\mathcal{K}^{op},Set] \longrightarrow \mathcal{K}$ has a further left adjoint $W$.

$$(W \dashv X \dashv Y) \;\colon\; \mathcal{K} \stackrel{\overset{W}{\hookrightarrow}}{\stackrel{\overset{X}{\leftarrow}}{\underset{Y}{\hookrightarrow}}} PSh(\mathcal{K}) \,.$$

Properties

If $\mathcal{K}$ is totally distributive, then since $Y$ is fully faithful, then, by the properties of adjoint triples, so is $W$. Thus, $\mathcal{K}$ is a coreflective subcategory of $[\mathcal{K}^{op},Set]$, which is cototal (or more precisely, pro-cototal?, since it is not locally small) — hence $\mathcal{K}$ is also cototal.

Moreover this means that the induced adjoint pair of (co-)monads

$$W X \dashv Y X \;\colon\; PSh(\mathcal{K}) \leftrightarrow PSh(\mathcal{K})$$

is an adjoint modality.

Every totally distributive category is an example of a completely distributive category.

Examples

• If $C$ is a small category, then its presheaf category $[C^{op},Set]$ is totally distributive.

• If $\mathcal{K}$ is totally distributive and $\mathcal{L}\subseteq \mathcal{K}$ is a full subcategory that is both reflective and coreflective, then $\mathcal{L}$ is totally distributive.

Waves

For any total category, it is possible to define the functor $W:\mathcal{K} \to [\mathcal{K}^{op},Set]$ which “wants to be” the left adjoint of $X$. The elements of $W(A)(K)$ are called waves from $K$ to $A$ (just because they are usually denoted by wavy arrows), and as in the case of continuous categories they form an idempotent comonad on $\mathcal{K}$ in the bicategory of profunctors. A total category is totally distributive if and only if $W \dashv X$.

Related pages

• distributive category

• completely distributive category

• Totally distributive categories are “almost” an example of continuous algebras for a lax-idempotent 2-monad.

References

• Robert Rosebrugh, Richard J. Wood, An adjoint characterization of the category of sets. PAMS 122 2 (1994) 409-413 [jstor:2161031]

• Rory Lucyshyn-Wright, Totally distributive toposes. spnet.

• Francisco Marmolejo, Bob Rosebrugh, and Richard Wood, Completely and totally distributive categories I, JPAA 216 no. 8-9 (2012). PDF

• Richard J. Wood, “The waves of a total category.” Theory and Applications of Categories 30.47 (2015): 1624-1646.