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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
    Commented Apr 8, 2021 at 22:29
  • $\begingroup$ @ZhenLin This is great; thanks! $\endgroup$
    – Emily
    Commented Apr 8, 2021 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
    Commented Apr 9, 2021 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
    Commented Apr 15, 2021 at 23:38

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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.

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    $\begingroup$ 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
    Commented Apr 16, 2021 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
    Commented Apr 16, 2021 at 12:59
  • $\begingroup$ Note that the construction Simon mentions is fairly standard; it's called an "internal full subcategory" and is mentioned for instance in B2.3.5 of Sketches of an Elephant. If the reason it feels unsatisfactory is that the internal categories have to be "completed" or "Yoneda-ified" into locally internal ones before you can take their union, you may be interested in the "variation through enrichment" perspective that identifies locally internal categories with categories enriched in a bicategory of spans, in which case internal categories are precisely the one-object ones. $\endgroup$
    – Mike Shulman
    Commented May 27, 2022 at 10:13
  • $\begingroup$ The original paper is Variation through enrichment by Betti, Carboni, Street, and Walters. There's also a more general version which I recently exposited that deals with fiberwise enrichment in an indexed monoidal category. $\endgroup$
    – Mike Shulman
    Commented May 27, 2022 at 10:15

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