Topoi and logoi | Higher Topos Theory notes

4.1. Topoi and logoi

Following Anel and Joyal (2021), we use separate words for the two directions of the adjunction underlying a geometric morphism: topos for the geometric direction and logos for the algebraic direction. Their terminology emphasizes a duality analogous to that between affine schemes and commutative rings. It is not yet common, and Marc did not use it in his lectures, but it makes the variance of the constructions below more transparent.

Definition 4.1.

Given two topoi $$S$$ and $$T$$ , a morphism of topoi (or geometric morphism) from $$S$$ to $$T$$ is a functor $\phi_*\colon S \to T$ which admits a left exact colimit-preserving left adjoint $\phi^*\colon T \to S$ . We denote by

\Topos \quad \subseteq \quad \Cat

the subcategory spanned by the topoi and the morphisms of topoi.

Definition 4.2.

A category $$T$$ is called a logos if it is a topos, i.e. if it is presentable and satisfies descent for all colimits. Given two logoi $$S$$ and $$T$$ , a morphism of logoi (sometimes called algebraic morphism\footnote{We warn the reader that this terminology clashes with the notion of `algebraic morphism' used by Lurie (2009, Definition 6.3.6.1).}) from $$S$$ to $$T$$ is a left exact colimit-preserving functor $\phi^*\colon S \longrightarrow T$ . We denote by

\Logos \quad \subseteq \quad \Cat

the subcategory spanned by the logoi and the morphisms of logoi.

Remark 4.3.

By Lemma 3.8, every morphism of logoi preserves $$n$$ -truncated objects and commutes with $$n$$ -truncations for all $$n$$ .

Remark 4.4.

By the adjoint functor theorem, every morphism of logoi $\phi^*\colon S \to T$ admits a right adjoint $\phi_*\colon T \to S$ , which is then a morphism of topoi. This assignment can be made functorial: there is an equivalence of categories

\Topos \quad \simeq \quad \Logos\catop, \qquad T \mapsto T, \qquad \phi_* \mapsto \phi^*.

More precisely, this equivalence arises as a restriction of the equivalence $\PrR \simeq (\PrL)\catop$ from [Lurie 2009, Corollary 5.5.3.4].

We now upgrade $\Logos$ and $\Topos$ to 2-categories.

Construction 4.5.

The category $\Cat$ is cartesian closed (the functor $C \times -\colon \Cat \to \Cat$ admits a right adjoint $\Fun(C,-)\colon \Cat \to \Cat$ for all $$C$$ ) and thus enriched over itself. This gives rise to a 2-category $\bbCat$ , whose objects are categories, whose morphisms are functors, and whose 2-morphisms are natural transformations between functors: $\Hom_{\bbCat}(C,D) = \Fun(C,D)$ .

This in particular allows us to upgrade the category $\Logos$ to a 2-category, by regarding it as a locally full subcategory of $\bbCat$ :

\bbLog \quad \subseteq \quad \bbCat.

The Hom-categories $\bbLog$ are given by the full subcategories

\Fun_{\bbLog}(S,T) \quad \subseteq \quad \Fun(S,T)

spanned by the morphisms of logoi. We may similarly upgrade $\Topos$ to a $$2$$ -category, by setting

\bbTop \quad := \quad \bbLog\catop \quad \subseteq \quad \bbCat\catop.

In other words, we define the Hom-categories of $\bbTop$ to be $\Fun_{\bbTop}(T,S) := \Fun_{\bbLog}(S,T)$ .

Remark 4.6.

The subcategory $\Fun_{\bbLog}(S,T)$ of $\Fun(S,T)$ is denoted by $\Fun^*(S,T)$ by Lurie. Marc in his lectures denoted $\Fun_{\bbTop}(T,S)$ by $\Geom(T,S)$ .

A point in a topological space $$X$$ is the same as a continuous map $* \to X$ from the one-point space, i.e. the terminal object in the category of topological spaces. Given that the terminal object of $\Topos$ is $\An$ (see Example 4.12 below), we get the following analogous notion for topoi:

Definition 4.7.

A point of a topos $$T$$ is a geometric morphism $\An \to T$ . We write

\Pt(T) \quad := \quad \Fun_{\bbTop}(\An,T)

for the category of points.

Example 4.8.

Let $T = \PSh(C)[\Sigma^{-1}]$ be a left exact localization of a presheaf category. Then the inclusion

\Pt(T) \quad \simeq \quad \Fun_{\bbLog}(T,\An) \quad \hookrightarrow \quad \Fun_{\bbLog}(\PSh(C),\An) \quad \simeq \quad \Pro(C)\catop

identifies $\Pt(T)$ with the $\Sigma$ -local pro-objects in $$C$$ , i.e. those formal systems $(X_i)_{i \in I}$ for which the functor

\PSh(C) \to \An, \qquad F \longmapsto \colim_i F(X_i)

inverts all morphisms in $\Sigma$ . Here we use the standard correspondence between pro-objects of $$C$$ and left exact colimit-preserving functors $\PSh(C)\to\An$ : a pro-object $$(X_i)_i$$ determines the displayed functor, and every such functor arises uniquely in this way.

References

Mathieu Anel, André Joyal. Topo-logie. In New spaces in mathematics. Formal and conceptual reflections, 155–257, Cambridge: Cambridge University Press. 2021.
Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.