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Stream: learning: questions

Topic: Theorem 5.6.5 in Category Theory in Context

Bernd Losert (Feb 03 2025 at 20:16):

Part (ii) of this theorem says that monadic functors create colimits that are preserved by the monad T and its square T². But if T preserves colimits, so does T² right? Why does it explicitly need to mention that T² must also preserve colimits?

Mike Shulman (Feb 03 2025 at 20:19):

I agree, that should be automatic.

Ryuya Hora (Feb 04 2025 at 02:03):

I remember that I had the exact same question while reading the book. I agree that it is automatic in most contexts, but I feel there is a reason to explicitly mention $T^2$, unless we only consider "all colimits of fixed shapes".

For example, if $T$ preserves all pushouts, then $T^2$ automatically preserves all pushouts. However, even if we assume that $T$ preserves a specific pushout $A \leftarrow B \rightarrow C$, there is no reason for $T$ to preserve the induced pushout $TA \leftarrow TB \rightarrow TC$.

I don't know of any illuminating examples of this subtlety.

Mike Shulman (Feb 04 2025 at 02:27):

Ah, I see -- I didn't look at the reference, but if it's talking about specific colimits rather than general shapes of colimits then I see it. It is pretty unusual to do the latter, but you could.

Nathanael Arkor (Feb 04 2025 at 05:20):

I've found this wording confusing in the past. I think a clearer equivalent condition is:

$T$ preserves the colimit of $F$ and the colimit of $TF$