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Stream: learning: questions

Topic: Classification theory for accessible categories

Jan Pax (Dec 24 2023 at 13:19):

could please someone explain to me on the page 7 here
why there are such morphisms $f',g',u,v$
and why $F$ is $id_{\cal C}$

James Deikun (Dec 29 2023 at 13:51):

$F$ is not $\mathrm{id}_{\mathcal{C}}$, it just restricts to it on the image of the inclusion of $\mathcal{C}$ into $\mathrm{Ind}(\mathcal{C})$ -- this is a background assumption being brought in, that we are using the standard functor $F : \mathrm{Ind}(\mathcal{C}) \to \mathcal{K}$ that takes each Ind-object to the corresponding colimit in $\mathcal{K}$.

James Deikun (Dec 29 2023 at 13:52):

The morphisms, on the other hand, exist because $\mathcal{K}$ is $\lambda$-accessible and so (classically) there must be morphisms (not necessarily unique) between objects of $\mathcal{C}$ witnessing "where the difference comes from" for any two different morphisms of $\mathcal{K}$. To construct them basically you take $K$ and $L$ and present them as $\lambda$-directed colimits of objects in $\mathcal{C}$, then first off $f,g$ correspond to cocones into $L$. Since they are different cocones, pick a position in them where the morphisms into $L$ are different; the object that is the source of those morphisms is $A$, and call the morphisms $f'',g''$. The corresponding morphism in the colimiting cocone for $K$ is $u$.

James Deikun (Dec 29 2023 at 13:52):

Then we have two morphisms from $A$, a $\lambda$-presentable object, to $L$, a $\lambda$-directed colimit. Thus $f'',g''$ factor essentially uniquely through two coprojections as $l_{\alpha_f} \circ f''', l_{\alpha_g} \circ g'''$. Because $\lambda$-directed colimits are, in particular, directed, there is a common upper bound $\alpha_{f,g}$ for the indices $\alpha_{f},\alpha_{g}$. For $B$ take $L_{\alpha_{f,g}}$, for $v$ the coprojection $l_{\alpha_{f,g}}$, and for $f',g'$ take $L_{\alpha_f \le \alpha_{f,g}} \circ f''', L_{\alpha_g \le \alpha_{f,g}} \circ g'''$.