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$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Nat}{\mathrm{Nat}}$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:

  1. 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.

  2. 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?

  3. 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.)

  4. 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?

  5. 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.

  6. 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 $V$.

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  • $\begingroup$ I really have absolutely no clue about (1), but two papers come to my mind: "Duality and small functors" and "A classification of small linear functors". $\endgroup$ Commented Apr 15 at 21:56
  • $\begingroup$ @IvanDiLiberti (and Emily): I would say a way to make precise the question is: do Spec and O form a Quillen adjunction? 100% sure no one claimed it before [but it might be someone did the exercise just for fun], but the proof might be just a matter of checking carefully where trivial co/fibrations go... $\endgroup$
    – fosco
    Commented Apr 16 at 6:26
  • $\begingroup$ @fosco I think how interesting such a result might be would depend a lot on the specific model structure one puts on $\mathsf{cSets}^\mathsf{op}$. One of the things I'm wondering about in this direction, but which is too far out of the scope of the current question, is if there's a kind of homotopy theory for “$\infty$-cogroupoids” dual to the one for $\infty$-groupoids (relevant MO question), and then, if there is one, how does Isbell duality relate both homotopy theories (if at all)? $\endgroup$
    – Emily
    Commented Apr 16 at 21:35
  • $\begingroup$ @fosco: What model structure do you have in mind for cosimplicial sets? Are there any such model structures in the literature? Certainly, one has a nontrivial class of weak equivalences (natural transformations whose homotopy limits in the ∞-category of spaces are equivalences). $\endgroup$ Commented Apr 19 at 18:11
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    $\begingroup$ Here is a trivial observation about the functor $\sf O$: it sends all simplicial spheres $S^n=Δ^n/∂Δ^n$ ($n>0$) as well as $Δ^0$ to the same cosimplicial set. Thus, $\sf O$ destroys a lot (most?) information in a simplicial set. $\endgroup$ Commented Apr 20 at 17:50

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