Assuming the Grothendieck axiom of universes (see also here or here), let $U_0$ denote the universe of "ZFC sets", i.e. of sets that can be constructed in ZFC alone without assuming axioms equivalent to the existence of inaccessable cardinals. Then the "standard" category of sets "$Set$" corresponds to all sets in $U_0$. (This paper calls $U_0$ "small-definable" sets.)
The "standard" Yoneda embedding is a functor $Y: \mathcal{C} \to Set^{\mathcal{C}^{op}}$ that takes a locally-$U_0$-small category $\mathcal{C}$ to its category of ($U_0$-)presheaves ($Set^{\mathcal{C}^{op}}$) by sending each object of $\mathcal{C}$ to its representable presheaf (and acting correspondingly on morphisms).
Given a Grothendieck universe $U$, let $U$-$0Cat$ denote the ($1$-)category of $U$-small $0$-categories. Then
Question: is it always possible to define a Yoneda embedding functor $Y: \mathcal{C} \to U\text{-}0Cat^{\mathcal{C}^{op}}$ for any locally-$U$-small category $\mathcal{C}$? In other words
(1) can we replace "sets" with "$0$-categories" in the statement of the Yoneda lemma? and
(2) does the Yoneda lemma still work for arbitrary Grothendieck universes $U$, rather than just $U_0$?
I assume from here that the answer to (2) is yes, so the answer (1) is more interesting to me (whence the question title).
Motivation:
- Size issues are apparently inherent to the notions of presheaf categories and the Yoneda embedding, see e.g. these answers (1)(2)(3) to questions from MathOverflow. In particular, although the "standard" statement gives the impression that the Yoneda Lemma can only possibly apply to categories $\mathcal{C}$ that are locally-$U_0$-small, it seems that in fact there are important applications to categories larger than this. (Cf. example 8.1 here.)
- The appearance of sets (which can be axiomatized/defined in many different ways) in one of the foundational/most important results/constructions in category theory always seemed bizarre/"artificial" to me. But a theorem relating the structure of ($1$-)categories to those of $0$-categories seems much more "natural" and much less "artificial".
Possibly misguided assumptions motivating question:
- Due to the "philosophy" of "negative thinking"/"decategorification"/"n-categories", an $(n-1)$-category should be equivalent (strictly? weakly?) to an $n$-category in which all $n$-morphisms are the identity. Thus a $0$-category should be "the same thing" as a $1$-category that is both discrete and skeletal.
- The category of all $U_0$-small, discrete, and skeletal ($1-$)categories is isomorphic to the category of all $U_0$-small ("ZFC-small") sets "$Set$", while the category of all essentially-$U_0$-small, discrete ($1-$)categories is equivalent to "$Set$". Cf. these two related questions (1)(2). (Although perhaps the latter category cannot exist due to size reasons; I don't know.)
- If the ($1$-)category $\mathcal{C}$ is $U_n$-small (thus in particular $U_n$-locally-small), the category of presheaves $U$-$0Cat^{\mathcal{C}^{op}}$ into which the ($U$-)Yoneda embedding embeds $\mathcal{C}$ is always $U_n$-large for any universe $U$, even for the $U_0$ ("ZFC") presheaf category $U_0$-$0Cat^{\mathcal{C}^{op}}$ (corresponding to the "standard" Yoneda embedding). At least that's what I inferred from these related questions (1)(2)(3) where one of the main conclusions is that the ("standard") Yoneda embedding is never essentially surjective due to size issues. (This answer seems to be related to the "uncheatable lemma" from here or here.) nLab also seems to imply something like this.
