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Stream: learning: questions

Topic: commutation of colim with hom

Jan Pax (Mar 13 2024 at 20:33):

In the snippet below I'm not sure about the first equivalence on the -4rd line, why colim commute with hom if $L$ is $$\lambda-$$presentable .Snímek-obrazovky-2024-03-13-213023.png

Kevin Carlson (aka Arlin) (Mar 13 2024 at 20:34):

The definition of $\lambda$-presentable is that its homs commute with $\lambda$-directed colimits.

Jan Pax (Mar 13 2024 at 20:40):

I'm aware only of this defintion:
Snímek-obrazovky-2024-03-13-214013.png

Kevin Carlson (aka Arlin) (Mar 13 2024 at 20:47):

That's not a definition of $\lambda$-presentable but of finitely presentable. Look a bit further down that chapter.

Jan Pax (Mar 13 2024 at 21:04):

My problem is not $\lambda$ vs. f.p. but rather how that bijection in -4rd line works, namely how can I choose that $i$ index suitably. The bijection doesn't seem to me to work for any $i$.

Reid Barton (Mar 13 2024 at 21:09):

All the colimits are over $i \in I$.

Reid Barton (Mar 13 2024 at 21:10):

That is, $i$ is a bound variable (by the colimit) in all of these expressions; these are not isomorphisms between objects that depend on $i$.

Jan Pax (Mar 14 2024 at 09:17):

But in my second snippet there they write: "if there exists $i$ such that ...." I'm somewhat uncertain how the $i$ comes into the play in the 4 isomorphisms in the first snippet :-( I've also thought about your remark that $i$ is bounded but still ??

Kevin Carlson (aka Arlin) (Mar 14 2024 at 16:20):

"If there exists $i$ such that..." is a way of spelling out what it actually means that $\mathrm{Hom}(x,\mathrm{colim} y_i)\to \mathrm{colim}\mathrm{Hom}(x,y_i)$ should be an isomorphism: every map out of $x$ into the colimit factors through some $y_i,$ and any two such factorizations are coequalized at some level in the diagram of the $y_i$.

Kevin Carlson (aka Arlin) (Mar 14 2024 at 16:22):

It's not clear what you mean by "how the $i$ comes into play". The $i$ is an index for the value of the functor $D$. We're taking the colimit over $i$ of various functors related to $D,$ such as $F\circ D$ and $i\mapsto \mathrm{hom}(L,F(D_i)).$