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Stream: learning: questions

Topic: 1-limits are 2-limits in an (op)representable category

Fernando Chu (May 26 2025 at 14:39):

The following is about strict 2-categories, functors, limits, etc. In a paper (p. 109), Street remarks

In a 2-category $K$ which is both representable and oprepresentable, $\Phi$ has a left 2-adjoint $\Psi$ and any limit which exists in $K_0$ is automatically a 2-limit in $K$.

Here representable means that $K$ has 2-pullbacks and all comma objects, and $\Phi$ is defined on objects by $\Phi(A):=1_A/1_A$, the comma object. From what I can tell, the remark is meant to follow from:

• The 2-adjunction $\Psi \dashv \Phi$.
• The isomorphism of categories $K(S,\Phi(A)) \cong \textbf{Cat}(\mathbb{2}, K(S,A))$, where $\mathbb{2}$ is the walking arrow.
• The 2-category of internal categories in a category with pullbacks is representable.

But I'm not sure what the argument is, could someone please elaborate on this? More generally, are there other nice conditions that imply that a 1-limit is also a 2-limit?

Mike Shulman (May 26 2025 at 15:01):

In general, for any enriching base $V$, if a $V$-category $C$ has copowers (aka tensors) by objects composing some strong generating family for $V$, then any limit in $C_0$ is also a conical limit in $C$, since then for each $G$ in the generators

$$V_0(G,C(X,\lim_i Y_i)) = C_0(G\odot X, \lim_i Y_i) = \lim_i C_0(G\odot X, Y_i) = \lim_i V_0(G, C(X, Y_i)) = V_0(G \lim_i C(X, Y_i))$$

hence (after verifying that this isomorphism is the one induced by the canonical map) $C(X, \lim_i Y_i) = \lim_i C(X,Y_i)$. In Cat, 1 and $\mathbf{2}$ form a strong generator, so copowers with those suffice.

Fernando Chu (May 26 2025 at 16:39):

Very clear explanation, thanks a lot!