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Stream: theory: category theory

Topic: cocontinuity of lax colimit embedding

Tom Hirschowitz (Aug 06 2024 at 13:06):

Sorry for spamming, but here's a second question in a row: if we consider the lax colimit $𝐀$ of a functor $F\colon 𝐂 → 𝐃$ in $𝐂𝐚𝐭$, we get embeddings $in₀\colon 𝐂 → 𝐀$ and $in₁\colon 𝐃 → 𝐀$, equipped with a natural transformation $λ\colon in₀ ∘ F → in₁$. If I'm not mistaken, the embedding $in₁\colon 𝐃 → 𝐀$ is always cocontinuous. Is this right, and if so do you know a reference for it?

Kevin Carlson (Aug 06 2024 at 21:09):

I guess it follows from the fact that $in_1$ is a fully faithful cosieve, right?

Kevin Carlson (Aug 06 2024 at 21:15):

I believe that $in_1$ is also a right adjoint, which is cool if I'm not missing something.

Tom Hirschowitz (Aug 06 2024 at 21:48):

Thanks, @Kevin Carlson, I'll look into this at some point but don't have time right now.

Tom Hirschowitz (Sep 18 2024 at 15:02):

Ok, @Yoann Barszezak and I finally took some time to look into this, sorry for the delay!
First of all, I swapped $in₀$ and $in₁$ in the type of $λ$, sorry about that too, I just fixed the original message.
Also, according to the nlab, a [[sieve]] is necessary fully faithful, so we omit to mention it.
Our conclusions are that:

• First of all, $in_1$ is a full coreflection, hence in particular a left adjoint, hence cocontinuous. This may be proved purely 2-categorically from the universal property of $𝐀$. In fact, it is also a sieve, hence it is continuous.
• Also, $in₀$ is indeed a cosieve, hence it is cocontinuous. This is also elementary, but we don't know how to prove it 2-categorically.

(However, we still don't have any reference...)
Thanks again for your help!