Every topos is cartesian closed: has finite products and exponentials. The product $\times$ and exponential $(-)^{(-)}$ make topos into closed monoidal category.

This enables lambda calculus and functional programming semantics in toposes.

The exponential $B^A$ is internal hom: morphisms $A \to B$ in topos correspond to global elements $1 \to B^A$. This internalizes function spaces.

For object $A$ in topos, subobjects $\text{Sub}(A)$ form Heyting algebra (intuitionistic logic lattice), not Boolean algebra in general. Operations:

• Meet: intersection
• Join: union
• Implication: Heyting implication
• Negation: pseudo-complement

In $\textbf{Set}$, this recovers Boolean algebra.

Category $\mathcal{E}$ is Grothendieck topos (sheaves on site) if and only if:

1. $\mathcal{E}$ has small colimits
2. Small colimits are universal
3. $\mathcal{E}$ has small generating family
4. Equivalence relations are effective

This characterizes sheaf toposes internally.

In topos, axiom of choice implies excluded middle: $P \vee \neg P$ for all propositions $P$. Conversely, if topos satisfies both axiom of choice and excluded middle, its internal logic is classical.

Most geometric toposes are intuitionistic, lacking choice and excluded middle.

A natural number object (NNO) in topos $\mathcal{E}$ is object $\mathbb{N}$ with $0: 1 \to \mathbb{N}$ and $s: \mathbb{N} \to \mathbb{N}$ (successor) such that for any $a: 1 \to A$ and $f: A \to A$, there exists unique $h: \mathbb{N} \to A$ with $h \circ 0 = a$ and $h \circ s = f \circ h$ (recursion).

Not all toposes have NNO, but those that do support arithmetic and recursion theory.