The n-Category Café

A follow up to the thought that ionads might come into the picture. We have Awodey and Kishida writing

The topological interpretation of this paper was originally formulated in terms of category and topos theory; this paper has served to reformulate it purely in terms of elementary (point-set) topology. In the original expression, we consider the geometric morphism from the topos $Sets/|X|$ of sets indexed over a set $|X|$ to the topos $Sh(X)$ of sheaves over a topological space induced by the (continuous) identity map $id: |X| \to X$. The modal operator $\Box$ is interpreted by the interior operation $int$ that the comonad $id^{\ast} \circ id_{\ast}$ induces on the Boolean algebra $Sub_{Sets/|X|}(id^{\ast} F) \cong \mathcal{P}(F)$ of subsets of $F$.(p.21)

Although the topological formulation presented here is more elementary and perspicuous, the topos-theoretic one in more useful for generalizations. For example, we see from it that any geometric morphism of toposes (not just $id^{\ast} \dashv id_{\ast}$) induces a modality on its domain. This immediately suggests natural models for intuitionistic modal logic, typed modal logic, and higher-order modal logic. (p.22)

Meanwhile in Garner’s paper, an ionad is given by a set $X$ of points together with a cartesian (i.e., finite limit preserving) comonad $I_X: Set^X \to Set^X$. The category of opens $O(X)$ of an ionad $X$ is the category of $I_X$-coalgebras.

…given any surjective geometric morphism $f: Set^X \to \mathcal{E}$, we obtain an ionad $(X, f^{\ast}f_{\ast})$ whose category of open sets is equivalent (by surjectivity of $f$) to $\mathcal{E}$. (p. 5)

David wrote:

I don’t suppose ionads could figure in a first-order modal logic.

Should we interpret that as

I do suppose ionads could perhaps figure in a first-order modal logic

? Otherwise it seems a strange thing to mention.

(I ask as a fellow Englishman…)

item (1) (Hilken and Rydeheard) is from 1999, I believe. The question of completeness posed at the end of section 4 is answered by the completeness result in the following paper:

Topology and modality: The topological interpretation of first-order modal logic. S. Awodey and K. Kishida, The Review of Symbolic Logic, 2008.

which is here:

http://www.andrew.cmu.edu/user/awodey/preprints/FoS4.phil.pdf