Phenomena of topos

Question

These days I am wandering on a wild adventure in an incredible but intimidating land. Fortunately, I could find a guide to some animals of this land

Phenomena of gerbes

But someone said to me that this wild land was crowded with topos. I thought collecting examples of topos you found in nature would be a good idea for my stay here. So my question is

What is your favorite example of (Grothendieck) topos?

I can start citing the simplest ones as examples. The category of sheaves of sets on a topological space (in particular the category of sets) and the category of $G$-sets (i.e. sets equipped with a left action of $G$).

It would be preferable if you give, along with the definition, a simple explanation of its context.

Answer 1

I like the example of the non-Grothendieck $\mathbf{Set}$-topos (i.e. there is a non-bounded geometric morphism to $\mathbf{Set}$) given by the category of continuous actions of a proper class-sized topological group. Since this is foundationally a bit subtle, either you take a Grothendieck universe, and work relative to that, or otherwise one can carefully write out a definition or even a characterisation of the underlying category without having to resort to having such data as a topology on a proper class. For instance, one could look at the axiomatisation of what it means for a topos to be the category of continuous actions of a topological group, but then remove the requirement for a generating set.

But other more ad-hoc constructions are sometimes possible. For example, if the topological group is the limit indexed by $\mathrm{ORD}$ of a sequence of open surjections between topological groups, then there's a very nice description in terms of a colimit of the underlying lex cocomplete categories of the continuous actions of these topological groups, and inverse image functors.

Answer 2

Let $A$ be a commutative ring. Let $\mathcal{E}$ be the category of finite étale $A$-algebras, i.e. those $A$-algebras $B$ such that

1. $B$ is finitely generated and projective as an $A$-module
2. The kernel $I$ of the multiplication map $\mu\colon B\otimes_AB\to B$ satisfies $I^2=I$.

Then $\mathcal{E}^{\text{op}}$ is an elementary boolean topos (but not a Grothendieck topos, as there are no infinite coproducts). The standard way to prove this is by a long detour into algebraic geometry, but it is a nice example to do it more directly. We can take $A$ to be a field or the ring of integers in a number field, and many facts of Galois theory and algebraic number theory have natural interpretations in terms of the corresponding topoi.

Answer 3

My favorite topos is a presheaf topos: the category of simplicial sets.

Answer 4

Terence Tao's cheap non-standard analysis can be interpreted as happening in a certain elementary topos, which I define here. Amusingly, this construction can itself be interpreted as happening in the category of sheaves on a certain space, but I digress.

Let $A$ be a set and let $F$ be a filter of $P (A)$. We define $\textbf{Set}^A / F$ to be the following category:

• The objects are indexed families of sets where the indexing set is in $F$.

• The morphisms $X \to Y$ are equivalence classes of indexed families of maps $f$ where:

  • The indexing set is in $F$.
  • For each index $a$ where $f_a$ is defined, $X_a$ and $Y_a$ are also defined and $f_a$ is a map $X_a \to Y_a$.
  • Two families are equivalent if the set of indices where they coincide is in $F$.

• Composition and identities are inherited from $\textbf{Set}$.

Then $\textbf{Set}^A / F$ is an elementary topos: finite limits and power objects can basically be constructed componentwise. In particular, the diagonal embedding $\Delta : \textbf{Set} \to \textbf{Set}^A / F$ is a logical functor. Since natural numbers objects are preserved by logical functors, it follows that $\textbf{Set}^A / F$ has a natural numbers object: $\Delta (\mathbb{N})$. Furthermore, $\textbf{Set}^A / F$ is boolean and satisfies the external axiom of choice. Thus, $\textbf{Set}^A / F$ satisfies the axioms of ETCS except possibly for well-pointedness.

It is not hard to see that the lattice of subobjects of $1$ in $\textbf{Set}^A / F$ is isomorphic to the boolean algebra $P (A) / F$. In particular, if $F$ is not an ultrafilter, then $\textbf{Set}^A / F$ is not well-pointed: it will have subterminal objects that are neither $0$ nor $1$. Conversely, if $F$ is an ultrafilter then $\textbf{Set}^A / F$ is well-pointed – so we obtain a non-standard model of ETCS.

In general, $\Delta : \textbf{Set} \to \textbf{Set}^A / F$ fails to have a right adjoint. Indeed, if $F$ is any non-principal filter, then $\Delta (F)$ is not the coproduct of its standard elements. Let $\mu_a : F \to 2$ be defined as follows: $$\mu_a (B) = \begin{cases} 0 & \text{if } a \notin B \\ 1 & \text{if } a \in B \end{cases}$$ Then, for each $B \in F$, $\mu (B)$ is equal to $1$ as morphisms $\Delta (1) \to \Delta (2)$. On the other hand, $\{ a \in A : \mu_a = 1 \} = \bigcap_{B \in F} B \notin F$, so $\mu$ is distinct from the constant $1$ as morphisms $\Delta (F) \to \Delta (2)$. Hence, the standard elements $\Delta (1) \to \Delta (F)$ do not comprise a jointly epimorphic cocone and cannot be a coproduct cocone.

(I think $\textbf{Set}^A / F$ also fails to be cocomplete in general, but the argument escapes me. Certainly this is so when $F$ is not closed under countable intersections – then $\Delta (\mathbb{N})$ fails to be the coproduct of its standard elements, but in an elementary topos, if the coproduct of countably many copies of $1$ exists, then the natural numbers object is the coproduct of the standard numerals.)