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Since the Yoneda lemma is so important, I'm curious what happens if you iteratively take the Yoneda embedding: Let $\mathcal{C}_0 = \mathcal{C}$ be a locally small category, and define $\mathcal{C}_{i+1}$ for $i\geq 0$ to be the functor category $\mathrm{Set}^{\mathcal{C}_i}$. This gives you a sequence of Yoneda embeddings

$$\mathcal{C}_0 \xrightarrow{\;\;j_0\;\;} \mathcal{C}_1 \xrightarrow{\;\;j_1\;\;} \dotsb \xrightarrow{\;\;\;\;} \mathcal{C}_i \xrightarrow{\;\;j_i\;\;} \dotsb$$

where the images of each $j_i$ is $\mathcal{C}_{i+1}$. There are a few questions you could ask here, for example

  • For which categories does this eventually stabilize? Like for which categories $\mathcal{C}$ do we eventually get an equivalence of categories between $\mathcal{C}_i$ and $\mathrm{Set}^{\mathcal{C}_i}$? This is a more general version of this question.

I've honestly thought very little about this, and am mostly curious if someone has written about this before and am looking for a reference/resource. But if anyone knows about this off the top of their head, or just wants to figure it out, feel free to make this page the reference. :)

Mike Pierce
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$\widehat{-} := [(-)^{op},\mathrm{Set}]$ is a 2-functor on $\mathrm{Cat}$, modulo (serious) size issues. It's actually a 2-monad: the unit is the Yoneda embedding $y_C:C \to \widehat{C}$, and the multiplication $\mu_C:\widehat{\widehat{C}}\to \widehat{C}$ is given by left Kan extension $$\mu_C(\Omega) := \mathrm{Lan}_{y_\widehat{C}}id_{\widehat{C}}(\Omega) = \int^{X\in \widehat{C}} X\cdot \Omega(X).$$

Given any category $C$, the presheaf category $\widehat{C}$ is a free algebra of this 2-monad. We can then form the (2-) bar construction, which gives a simplicial object $B(\widehat{-},\widehat{C}):\Delta\to \mathrm{Cat}$. I'm not aware of what has been written about this.

However, an equivalence $C\simeq \widehat{C}$ is not possible because of size issues. For a simpler reason than Cantor's argument that there can be no bijection $X\simeq P(X)$: if $C\in \mathrm{Cat}$ ($C$ is small, with a set of objects and a set of morphisms), then $\widehat{C}$ is large. Similarly if $C$ is only locally small; the 2-functor strictly increases the "universe level" each time, so there can be no equivalence. This is somewhat problematic, because $\widehat{-}$ is such an important map. The issue has been handled in terms of relative pseudomonads. Perhaps the question could be reformulated and answered in that context.

Christian Williams
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