In enriched category theory over a base monoidal category $(\mathcal{V},\otimes_{\mathcal{V}},\mathbf{1}_{\mathcal{V}})$, one can consider $\mathcal{V}$ itself as a $\mathcal{V}$-enriched category when it has a closed monoidal structure $[-,-]_{\mathcal{V}}$.

Is there a similar procedure in internal category theory? That is, starting from a category $(\mathcal{E},\times_{\mathcal{E}},\mathbf{1}_{\mathcal{E}})$ with pullbacks and a terminal object, can one associate an $\mathcal{E}$-internal category to $\mathcal{E}$ itself?

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    $\begingroup$ Yes and no. Think about the most fundamental category, $\textbf{Set}$: an internal category is a small category, and $\textbf{Set}$ is (usually) not even essentially small. But there is a notion of locally internal category and when you have a locally cartesian closed category it can be locally self-internalised. $\endgroup$ – Zhen Lin Apr 8 at 22:29
  • $\begingroup$ @ZhenLin This is great; thanks! $\endgroup$ – Théo Apr 8 at 22:51
  • $\begingroup$ @ZhenLin I found your comment edifying, thank you; would you mind posting it as an answer to close out the question? $\endgroup$ – Alec Rhea Apr 9 at 2:00
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    $\begingroup$ @ZhenLin You post so many good comments, most of them already answer the question. It is kind of sad that you don't post them as answers. This leaves the questions looking "unanswered", even when they are answered. $\endgroup$ – Martin Brandenburg Apr 15 at 23:38

Internal categories are too limited for self-internalisation. Think of the most fundamental category, $\textbf{Set}$: an internal category is a small category, and $\textbf{Set}$ is (usually) not even essentially small. (In NF and related set theories with a universal set, $\textbf{Set}$ has a set of objects but fails to be cartesian closed, so I think even then it cannot be self-internalised.)

However, there is a notion of locally internal category – a special kind of fibred (= indexed) category – and the codomain fibration (= self-indexing) of a category with finite limits is locally internal if and only if it is locally cartesian closed.

One thing that bugs me is that there is one feature of locally small categories that does not seem to be captured by locally internally categories, namely that any locally small category is a union of small categories. But I am not sure whether there is any merit in trying to formalise this in the language of fibred categories.

  • $\begingroup$ Regarding your last paragraph, what doesn't work with the following ? : Given a locally internal category $D$, encoded by a fibration $\overline{D} \to C$, for any "$C$-familly object" in $D$, i.e. an object $d \in \overline{D}$ over $c \in C$, you can construct an internal (small) category whose object of object is $c$, and which is a full subcategory of $D$. And $D$ is the union of these internal cateogry in the sense that any "object" of $D$ belong to one of these... Of course the union is externally indexed, but this the same with Set where the union is class indexed. $\endgroup$ – Simon Henry Apr 16 at 12:42
  • $\begingroup$ Yes, that’s true, but also somehow not satisfactory. I haven’t thought about it much, however. Maybe that is indeed enough. $\endgroup$ – Zhen Lin Apr 16 at 12:59

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