Todd Trimble Cartesian closure of internal categories

Proof

First suppose $\mathbf{E}$ is finitely complete. Then the category of directed graphs $\mathbf{E}^{\bullet \stackrel{\to}{\to} \bullet}$ is also finitely complete, and since $Cat(\mathbf{E})$ is monadic over $\mathbf{E}^{\bullet \stackrel{\to}{\to} \bullet}$, it follows that $Cat(\mathbf{E})$ is also finitely complete.

Now suppose that $\mathbf{E}$ is finitely complete and cartesian closed. Let $\Delta_3$ denote the category of nonempty ordinals up to and including the ordinal with 4 elements. We have a full and faithful embedding

where the codomain category is cartesian closed. Indeed, the exponential of two objects $F$, $G$ in $\mathbf{E}^{\Delta_3^{op}}$ may be computed as an $\mathbf{E}$-enriched end

when evaluated at $m \in Ob(\mathbf{E}^{\Delta_3^{op}})$, as is easily checked (see for instance here); note that this end is a finite limit diagram since $\Delta_3$ is finite.

If $C$, $D$ are internal categories in $\mathbf{E}$, seen as functors $\Delta_3^{op} \to \mathbf{E}$, the exponential $N C^{N D}$ defines the exponential in $Cat(\mathbf{E})$. To see this, it suffices to check that $N C^{N D}$, as defined by the end formula above, is a category $B$, i.e., is in the essential image of the nerve functor. For in that case, we have natural isomorphisms

whence $B$ satisfies the universal property required of an exponential.

Objects in the essential image of the nerve $N$ are characterized as functors $\Delta_3^{op} \to \mathbf{E}$ which take intervalic joins in $\Delta_3$ to pullbacks in $\mathbf{E}$, as given precisely by the Segal conditions. The remainder of the proof is then finished by the following lemma.

Proof

For any $X$ we have the formula

Since the enriched end and the internal hom-functor $Hom(X k, -)$ both preserve pullbacks, we are reduced to checking that

• If $C$ satisfies the Segal conditions, then so does
  $$\prod_{f \colon n \to k} \prod_{g \colon n \to m} C(n)$$

  as a functor $\Delta_3^{op} \to \mathbf{E}$ in the argument $m$ (for each fixed $k$).

Note that the displayed statement is a proposition in the language of finitely complete categories (i.e., in finitary essentially algebraic logic). Since hom-functors $\mathbf{E}(e, -) \colon \mathbf{E} \to Set$ jointly preserve and reflect the validity of such propositions, it suffices to prove it for the case where $\mathbf{E} = Set$. But this is classical elementary category theory; it says precisely that if $C$ is a small (ordinary) category, then the usual functor categories $C^{\mathbf{2}}$, $C^{\mathbf{3}}$ are equivalently described by exponentials of (truncated) simplicial sets. This completes the proof.