Interior spaces and frames

Today I attended a great talk by Ivan Di Liberti at YaMCATS and I relearned a nice little fact about interior spaces that Ivan showed me already some time ago. Let me write it down here before I forget again.

Ever since the beginning of topology, it was known that topological spaces can be equivalently represented as interior spaces, which are sets $X$ equipped with an interior operator $\mathrm{int}\colon \mathcal P(X) \to \mathcal P(X)$. The set $\mathcal P(X)$ is the powerset of $X$ and the interior operator needs to satisfy the following axioms:

• $\mathrm{int}(M) \subseteq M$
• $M \subseteq N$ implies $\mathrm{int}(M) \subseteq \mathrm{int}(N)$
• $\mathrm{int}(\mathrm{int}(M)) = \mathrm{int}(M)$
• $\mathrm{int}(M \cap N) = \mathrm{int}(M) \cap \mathrm{int}(N)$ and $\mathrm{int}(X) = X$

These look almost like the axioms of nuclei, except that the order in the first item is reversed and we have the extra assumption that $\mathrm{int}(X) = X$.

I am sure many people are familiar with with the adjunction between topological spaces and frames. However, what I haven’t seen before is the explicit construction of the adjunction between interior spaces and frames, and it’s rather nice!

Let us first show the mapping from frames to interior spaces. Recall that points $\mathrm{pt}(L)$ of a frame $L$ can be represented as frame homomorphisms $L \to 2$, where $2$ is the two element frame. Take the evaluation function: $$L \times \mathrm{pt}(L) \to 2,\quad (a,p\colon L\to 2) \mapsto p(a)$$ By currying, we have a function $$f\colon L \to 2^{\mathrm{pt}(L)} \cong \mathcal P(\mathrm{pt}(L))$$ It is a standard fact that this function is a frame homomorphism and it thus has a right adjoint $g$. Then, since $f$ preserves finite meets, the composition $f \circ g$ is an interior operator $\mathcal P(\mathrm{pt}(L)) \to \mathcal P(\mathrm{pt}(L))$ and so $(\mathrm{pt}(L), f \cdot g)$ is an interior space. So neat!

Conversely, given an interior space $(X,\mathrm{int})$, we simply take the frame $\tau$ of fixpoints of $\mathrm{int}$. The fact that this is well defined is precisely the same proof as when showing that $(X,\tau)$ is a topological space, in the correspondence between interior spaces and topological spaces.

Categorification

In fact, Ivan showed something much more sophisticated. Instead of frames he worked with Grothendieck toposes and instead of interior spaces he worked with certain (large) ionads, which are pairs $(\mathcal{X},\mathrm{Int})$ where $\mathcal{X}$ is a suitable locally small category and $\mathrm{Int}$ is a comonad on the full subcategory $\overline{\mathcal P}(\mathcal{X})$ of $[\mathcal{X},\mathrm{Set}]$, consisting of small copresheaves. The construction from Grothendieck toposes is basically as I described above (just replace $2$ with the category $\mathrm{Set}$) and to go from Ionads to toposes just take the category of coalgebras for the comonad $\mathrm{Int}$.

In fact, when specialised to interior spaces and frames, we obtain precisely the same construction as described above. For details, see Ivan’s older slides here or his Ph.D. thesis.