nLab
totally distributive category

Totally distributive categories

Totally distributive categories

Definition

Recall that a locally small category 𝒦\mathcal{K} is total if its Yoneda embedding Y:π’¦βŸΆ[𝒦 op,Set]Y\colon\mathcal{K}\longrightarrow [\mathcal{K}^{op},Set] has a left adjoint XX.

Definition

A total category 𝒦\mathcal{K} is totally distributive if X:[𝒦 op,Set]βŸΆπ’¦X \colon [\mathcal{K}^{op},Set] \longrightarrow \mathcal{K} has a further left adjoint WW.

(W⊣X⊣Y):𝒦β†ͺY←Xβ†ͺWPSh(𝒦). (W \dashv X \dashv Y) \;\colon\; \mathcal{K} \stackrel{\overset{W}{\hookrightarrow}}{\stackrel{\overset{X}{\leftarrow}}{\underset{Y}{\hookrightarrow}}} PSh(\mathcal{K}) \,.

Properties

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

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

WX⊣YX:PSh(𝒦)↔PSh(𝒦) W X \dashv Y X \;\colon\; PSh(\mathcal{K}) \leftrightarrow PSh(\mathcal{K})

is an adjoint modality.

Examples

Waves

By the adjoint functor theorem for total categories, a total category is totally distributive if and only if the functor XX preserves all limits. Moreover, for any total category, it is possible to define the functor W:𝒦→[𝒦 op,Set]W:\mathcal{K} \to [\mathcal{K}^{op},Set] which β€œwants to be” the left adjoint of XX (and is if 𝒦\mathcal{K} is total). The elements of W(A)(K)W(A)(K) are called waves from KK to AA (just because they are usually denoted by wavy arrows), and as in the case of continuous categories they form an idempotent comonad on 𝒦\mathcal{K} in the bicategory of profunctors.

References

  • R. Rosebrugh and R.J. Wood. An adjoint characterization of the category of sets. PAMS, Vol. 122, No. 2, 409–413, 1994.

  • 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

Last revised on November 16, 2018 at 13:37:12. See the history of this page for a list of all contributions to it.