The n-Category Café

First let’s prove that if (2) is true for all $C$, then (1) is true when $C$ and $I$ are small. Let $T$ be a monad on a small category $C$, let $\hat{C} = Set^{C^{op}}$ be the category of presheaves on $C$, and let $y\colon C\to \hat{C}$ be the Yoneda embedding. Since $\hat{C}$ is the free cocompletion of $C$, the monad $T$ extends uniquely to a cocontinuous monad $\hat{T}$ on $\hat{C}$. Saying that $\hat{T}$ “extends” $T$ means that $\hat{T}\circ y \cong y\circ T$, coherently with the monad structure. It follows that $T$-algebras can be identified with those $\hat{T}$-algebras whose underlying presheaves are representable, and we have the following diagram: $$\array{C^T & \overset{y^T}{\to} & \hat{C}^{\hat{T}}\\ ^u\downarrow && \downarrow^{\hat{u}}\\ C& \underset{y}{\to} & \hat{C}}$$ in which the horizontal functors are both fully faithful. Now let $d\colon I\to C^T$ be a diagram, with $I$ small, such that $u d$ has a limit $\ell$ in $C$. Since $y$ preserves limits, $y(\ell)$ is also a limit of $y u d \cong \hat{u} y^T d$ in $\hat{C}$. But $\hat{C}$ is complete, so by assumption, $\hat{C}^{\hat{T}}$ is complete and $\hat{u}$ is continuous. Therefore, since $I$ is small, $y^T d$ has a limit $\hat{\ell}$ in $\hat{C}^{\hat{T}}$ which is preserved by $\hat{u}$. This means that $\hat{u}(\hat{\ell})$ is a limit of $\hat{u} y^T d$, and hence isomorphic to $y(\ell)$. But then $\hat{\ell}$ is a $\hat{T}$-algebra whose underlying presheaf is representable (it is represented by $\ell$), and thus it “is” a $T$-algebra $k$ with $u(k)\cong \ell$. Finally, since fully faithful functors reflect limits, $k$ is a limit of $d$ which is preserved by $u$.

I actually find this argument kind of striking. It implies that once we’ve proven that (say) products and equalizers lift to the category of algebras for any monad, it follows automatically that all limits lift similarly—even when the base category doesn’t have products and equalizers out of which those other limits can be constructed! Of course, philosophers can debate what it means for two true statements to be equivalent, but in practice this sort of argument can simplify your life.

Now what about when $C$ is large? Let me first point out some things that don’t work. In general, without any universes, the category $Set^{C^{op}}$ will not exist. If $C$ is locally small, then the category of small presheaves on $C$ does always exist and is the free cocompletion of $C$, but it is not complete without further assumptions on $C$ (such as that $C$ is small, or itself complete). So it seems hard to get away without some universe-like hypotheses.

Of course, if we assume Grothendieck’s axiom of universes, then we can reason as follows: pick a universe $U$ such that $C$ and $I$ are $U$-small, and let $Set$ denote the $U$-large category of $U$-small sets; then the above argument goes through. Note that this requires the statement of (2) to be changed to “for any universe $U$ and any $C$ and $T$, if $C$ is $U$-small-complete, then so is $C^T$, and $u$ is $U$-small-continuous.” However, presumably whatever argument we originally used to prove it would still prove this more general statement.

On the other hand, in strong Feferman set theory ZMC/S, we have a specified universe $U$ which satisfies the reflection schema, and “small,” “large,” “complete,” and “continuous” have a fixed meaning referring to this $U$. We can now argue as follows: the above proof shows that (1) is true for any small categories $C$ and $I$. But this is just the “relativization” to $U$ of the statement (1) itself; thus by the reflection schema, (1) itself is true. Personally, I find this version cleaner, although (taking into account how one proves the consistency of Feferman set theory) they contain more or less the same content.