$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Nat}{\mathrm{Nat}}$Isbell duality sets up an adjunction (see here for a short abstract summary) $$\mathsf{O}\dashv\mathsf{Spec}\colon\mathrm{sSets}\rightleftarrows\mathrm{cSets}^\mathsf{op}$$ between simplicial sets and cosimplicial sets via \begin{align*} \mathsf{O}(X_\bullet)_n &= \Nat(X_\bullet,\Delta^n),\\ \mathsf{Spec}(Y^\bullet)_n &= \Nat(Y^\bullet,\Delta_n) \end{align*} for $X_\bullet$ a simplicial set, $Y_\bullet$ a cosimplicial set, and $\Delta_n$ the corepresentable cosimplicial set sending $[m]$ to $\mathrm{Hom}_{\mathbf{\Delta}}([n],[m])$.

**Question.** In general, what is known about Isbell duality in this setting? Has it been explored before somewhere in the literature?

For instance:

Is there any homotopy-theoretic significance to this adjunction?

a) Can we deduce information about $X_\bullet$ based on information about $\mathsf{O}(X)$, $[\mathsf{Spec}\circ\mathsf{O}](X)$, general properties of this adjunction, etc.?

b) What about the case where we take $X$ to be a Kan complex, a quasicategory, or an $(\infty,2)$-category as defined in Kerodon?

**Note 1:**As Dmitri notes, the functor $\mathsf{O}$ is very destructive, as it sends all simplicial spheres $\Delta^{n}/\partial\Delta^{n}$ as well as $\Delta^{0}$ to the same cosimplicial set.What do we know about Isbell self-dual simplicial sets, i.e. those for which the unit $X_\bullet\to[\mathsf{Spec}\circ\mathsf{O}](X_\bullet)$ is an isomorphism?

More generally when is the unit at a particular $X_\bullet$ an epimorphism?

(Motivation: $[\mathsf{Spec}\circ\mathsf{O}](X)_0$ is always just a point.)

Is there a good and nice description of the reflexive completion of $\mathbf{\Delta}$ in this case, i.e. the category of all Isbell self-dual simplicial sets?

The adjunction $\IsbellO\dashv\IsbellSpec$ defines a monad on $\mathsf{sSets}$. What are the algebras for this monad? Is there any homotopy-theoretic significance to them?

**Note 2:**By Theorem 2.7 of Ivan Di Liberti's*Codensity: Isbell duality, pro-objects, compactness and accessibility*, this monad is the same is the same as the codensity monad $\mathrm{Ran}_{\Delta^{(-)}}(\Delta^{(-)})$ of the Yoneda embedding $$\Delta^{(-)}\colon\mathbf{\Delta}\hookrightarrow\mathsf{sSets}$$ of $\mathbf{\Delta}$ given by $[n]\mapsto\Delta^{n}$.**Note 3:**$\mathsf{Vect}_k$-enriched Isbell duality for $\mathcal{C}$ the one-object $\mathsf{Vect}_{k}$-category associated to $k$ recovers the adjunction $\mathsf{Vect}_{k}\rightleftarrows\mathsf{Vect}^{\mathsf{op}}_{k}$ sending a vector space to its dual, the monad induced by this adjunction is isomorphic to the codensity monad of the inclusion $\mathsf{FinVect}_k\hookrightarrow\mathsf{Vect}_k$ (Leinster, Theorem 7.5), and the algebras for this monad are the linearly compact Hausdorff topological vector spaces (Leinster, Theorem 7.8).So I would guess the algebras for the monad induced by $\mathsf{O}\dashv\mathsf{Spec}\colon\mathrm{sSets}\rightleftarrows\mathrm{cSets}^\mathsf{op}$ might be closely related to simplicial compact Hausdorff topological spaces.

Given a simplicial set $X_\bullet$, there's a canonical map $$\mathrm{ev}_{X}\colon X_\bullet\boxtimes\mathsf{O}(X)^\bullet\to\mathrm{Tr}(\Delta)$$ from the functor tensor product of $X_\bullet$ with $\mathsf{O}(X)^\bullet$ to the trace of the simplex category, which is isomorphic to $\mathbb{N}_{\geq1}$ (see this other question). Can we deduce any interesting information about $X_\bullet$ from this map?

**Note 4:**For $\mathsf{Vect}_k$-enriched Isbell duality, this map takes the form $$\mathrm{ev}_{V}\colon V\otimes_k V^*\to k$$ and is given by $(v,\phi)\mapsto\phi(v)$, the*canonical duality pairing of $V$*.