Recently, I have some questions for some generalizations from algebraic topology.
I learn some homotopy theory in algebraic topology. We know that, if two spaces are homotopy, then they have same cohomology groups for constant sheaves.
I want to know if there are similar theory for topoi, such as homotopy theory for topoi?
Thanks for your answers
There is a notion of the étale homotopy type of a (Grothendieck) topos, going back to Artin and Mazur (I think). However, in classic "French" fashion they turned a theorem (in one setting) into a definition (in a more general setting): roughly speaking, two toposes have the same étale homotopy type if they have the same étale fundamental group and the same cohomology for all (constant sheaf) coefficients. So perhaps this isn't what you're looking for, but let me continue.
Let $\mathcal{E}$ be a topos. Artin and Mazur define the étale homotopy type of $\mathcal{E}$ under the assumption that $\mathcal{E}$ is locally connected, which for the purpose of this discussion means the unique colimit- and finite-limit-preserving functor $\Delta : \textbf{Set} \to \mathcal{E}$ has a left adjoint $\pi : \mathcal{E} \to \textbf{Set}$. Let $\textbf{Hc} (\mathcal{E})$ be the category of hypercovers of the terminal object $1$ in $\mathcal{E}$. (The objects are simplicial objects $K$ in $\mathcal{E}$ such that the unique morphism $K \to 1$ is a local trivial Kan fibration and the morphisms are simplicial homotopy classes of morphisms.) Let $\mathcal{K} = \operatorname{Ho} \textbf{sSet}$ be the category of simplicial sets localised with respect to weak homotopy equivalences. The étale homotopy type of $\mathcal{E}$, as defined by Artin and Mazur, is the following pro-object in $\mathcal{K}$: $$\Pi (\mathcal{E}) = \varprojlim_{K : \textbf{Hc} (\mathcal{E})} \pi (K)$$ Here, $\pi (K)$ denotes the simplicial set obtained by applying $\pi : \mathcal{E} \to \textbf{Set}$ degreewise to the simplicial object $K$. (The reason for using simplicial homotopy classes of morphisms in the definition of $\textbf{Hc} (\mathcal{E})$ is so that we get a cofiltered category. This category is usually not small, but it is coinitially small, so we do indeed get a pro-object.)
Observe that, for a Kan complex $X$, $$\textbf{Pro} (\mathcal{K}) (\Pi (\mathcal{E}), X) \cong \varinjlim_{K : \textbf{Hc} (\mathcal{E})^\textrm{op}} \mathcal{K} (\pi (K), X) \cong \varinjlim_{K : \textbf{Hc} (\mathcal{E})^\textrm{op}} \pi_0 [\textbf{s} \mathcal{E}] (K, \Delta (X))$$ where $\pi_0 [\textbf{s} \mathcal{E}]$ is the category of simplicial objects in $\mathcal{E}$ modulo simplicial homotopy. The simplicial analogue of Verdier's hypercovering theorem states that the RHS computes $\pi_0 (\textbf{R} \Gamma (\Delta (X)))$, which you might interpret as analogous to $H_0$ of the derived global sections of a chain complex of sheaves. This is literally the case when $X$ is a simplicial abelian group, so you can extract the classical Verdier hypercovering theorem from this. Put it another way, the pro-object $\Pi (\mathcal{E})$ encodes enough information to determine the cohomology of constant sheaves on $\mathcal{E}$, which is what you wanted – but as I said in the first paragraph, in some sense the definition was constructed to make this true.
Now let me discuss the notion of shape, which has been mentioned in the comments. The category $\mathcal{K}$ is not a pleasant category to work with, and $\textbf{Pro} (\mathcal{K})$ is even less pleasant. The restriction to locally connected $\mathcal{E}$ is also somewhat unsatisfying. In the years since Artin–Mazur new technologies have been developed for abstract homotopy theory and using this we obtain an improved version of the étale homotopy type.
First, we must obtain an $\infty$-topos from $\mathcal{E}$. Take the category of simplicial objects in $\mathcal{E}$ and localise (in the $(\infty, 1)$-categorical sense now) with respect to local weak homotopy equivalences to obtain an $\infty$-topos $\tilde{\mathcal{E}}$. Let $\mathcal{S}$ be the $(\infty, 1)$-category of $\infty$-groupoids. The global sections functor $\Gamma : \tilde{\mathcal{E}} \to \mathcal{S}$ has a left adjoint $\Delta : \mathcal{S} \to \tilde{\mathcal{E}}$, and as before we are interested in the composite functor $X \mapsto \pi_0 (\Gamma (\Delta (X)))$. Since $\Gamma$ is an accessible right adjoint and $\Delta$ is a left adjoint that preserves finite limits, the composite $\Gamma \Delta$ is an accessible functor that preserves finite limits. The $(\infty, 1)$-category of accessible functors $\mathcal{S} \to \mathcal{S}$ that preserve finite limits is equivalent to the $(\infty, 1)$-category of pro-objects in $\mathcal{S}$, so $\Gamma \Delta$ corresponds to some pro-object $\Pi (\tilde{\mathcal{E}})$ – the shape of $\tilde{\mathcal{E}}$ is this object. (Given an inverse diagram $T$ in $\mathcal{S}$, the functor $\varinjlim \mathcal{S} (T, -) : \mathcal{S} \to \mathcal{S}$ is an accessible functor that preserves finite limits; the fact is that all accessible functors $\mathcal{S} \to \mathcal{S}$ that preserve finite limits arise in this way.)
The advantage of the formulation of shape is that it is immediate how the shape encodes the cohomology of constant sheaves, but again this seems to be a trick – a different one from before, but in some sense there is still nothing deep going on. One might even say there is even less depth here because even hypercovers have disappeared from view. But perhaps this answers your question about whether there is a homotopy theory of toposes.
