Consider the object classifier of the $\infty$-topos of $\infty$-groupoids. For the role it plays in homotopy type theory as the type of types, let’s denote it as $Type = \coprod_{[F]} B Aut(F)$, the coproduct of the automorphism ∞-groups of all (small) homotopy types $[F]$.

Now attempting to imitate the duality between the category of sets and the category of complete atomic Boolean algebras, we might consider the map taking an $\infty$-groupoid $A$ to $[A, Type]$. Similarly to how the internal Boolean algebra structure on $\mathbf{2}$ induces a Boolean algebra structure on $[X, \mathbf{2}]$, for a set $X$, the internal $\infty$-topos structure on $Type$ induces an $\infty$-topos structure on $[A, Type] = \infty-Grpd/A$, the slice $\infty$-topos.

Question: Just as $[X, \mathbf{2}]$ is a special kind of Boolean algebra, being complete and atomic, how can I characterise those $\infty$-toposes of the form $[A, Type]$?

The answer will probably involve Mike Shulman's suggestions to me: having a set of tiny generators and being a Boolean presheaf $\infty$-topos.

Further questions: with such a characterisation in hand, should we expect a duality between such $\infty$-toposes and the $\infty$-topos of $\infty$-groupoids? What would be the equivalent of Stone Space-Boolean algebra duality?


1 Answer 1


Let $\mathcal{S}$ denote the $\infty$-category of spaces. For any $\infty$-topos $\mathcal{X}$, there is an essentially unique geometric morphism $\pi^{\ast}: \mathcal{S} \rightarrow \mathcal{X}$. The $\infty$-topos $\mathcal{X}$ has the form $\mathcal{S}_{/A}$ if and only if the geometric morphism $\pi^{\ast}$ is etale. This is true if and only if the following three assertions hold:

$(1)$ The functor $\pi^{\ast}$ admits a left adjoint $\pi_{!}$ (equivalently: $\pi^{\ast}$ preserves small limits).

$(2)$ The functor $\pi_{!}$ is conservative (that is, if $\alpha: X \rightarrow Y$ is a morphism in $\mathcal{X}$ for which $\pi_{!}(\alpha)$ is an equivalence, then $\alpha$ is an equivalence).

$(3)$ There is a projection formula $$\pi_{!}( \pi^{\ast} X \times_{ \pi^{\ast} Y } Z ) \simeq X \times_{Y} \pi_{!} Z.$$

The construction $A \mapsto \mathcal{S}_{/A}$ gives a fully faithful embedding from the $\infty$-category of spaces to the $\infty$-category of $\infty$-topoi. However, I wouldn't be inclined to see this as analogous to Stone duality: as you point out, Boolean algebras of the form $[X, \mathbf{2} ]$ are rather special. It's more analogous to the observation that any set can be regarded as a topological space by equipping it with the discrete topology.

To get a closer analogue of Stone duality, note that the $\infty$-category of $\infty$-topoi admits small limits. Consequently, the functor $A \mapsto \mathcal{S}_{/A}$ extends formally to a functor $F$ from Pro-spaces to $\infty$-topoi which commutes with filtered inverse limits. This functor $F$ is not fully faithful, but it is fully faithful when restricted to profinite spaces (that is, Pro-spaces which can be represented by filtered diagrams of spaces which have only finitely many homotopy groups, each of which is required to be finite). You therefore get an embedding $\theta:$ { Profinite spaces } $\hookrightarrow$ { $\infty$-topoi } which is a better analogue of Stone duality. In fact, it generalizes Stone duality: the RHS contains the ordinary category of sober topological spaces as a full subcategory, the LHS contains the ordinary category profinite sets as a full subcategory, and on the latter subcategory $\theta$ restricts to the usual fully faithful embedding { Profinite sets } $\simeq$ { Stone Spaces } $\hookrightarrow$ {sober topological spaces}.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.