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Aug
20
comment Locally presentable categories of categories
It seems a pity. Perhaps there is another way to do this, other than the process of fixing a $\kappa$?
Aug
20
comment Locally presentable categories of categories
I have spent some time asking around about this problem and I am guessing that categories like Group and Vec cover such wide sets (actually all sets) that you could not actually fix a $\kappa$ and find those categories as colimits. As you say, the colimit diagrams would get too large?
Aug
20
comment Locally presentable categories of categories
Can we choose a $\kappa$ such that we find categories like Group or Vect as colimits over diagrams of compact categories. I assume "compact" is relative to your choice of $\kappa$.
Aug
20
comment Locally presentable categories of categories
Hi, Thanks for responding. Yes, I am asking something quite vague, so perhaps I can modify the question with some help. Basically, I am trying to find a suitable category of categories that would support interesting categories as colimits. So, according to this link "Cat", which might be ambiguous, is lfp. You have suggested a modified Cat1 which is $\aleph_1$-presentable? The way I understand this, there is a plethora of flavours of $\kappa$-presentability for $\kappa$ all cardinals.
Aug
20
asked Locally presentable categories of categories
Aug
18
accepted Reconstruct a category : Forgetful fuctor to underlying graph, free functor of graph and then a quotient
Aug
18
asked Reconstruct a category : Forgetful fuctor to underlying graph, free functor of graph and then a quotient
Aug
9
asked Approximation to FRel in Cat
Aug
1
accepted sequence of colimit diagrams
Jul
30
asked sequence of colimit diagrams
Jul
17
asked Explain the compactness relation for elements of dcpos and also in a category if objects
Mar
7
awarded  Curious
Mar
4
awarded  Tumbleweed
Mar
2
comment Examples of colimits in a category of categories
Hi Ben, you have mentioned a motivating example. Before I was thinking about colimits in a category of categories, I was thinking about groups presented with generators and relations. In the category you talk about, we see that the finitely presentable categories form diagrams. These diagrams have colimits which are other groups that are not finitely presentable. The arrow between groups goes from one group to another group with the same axioms plus one or more extra axioms. Thus, the finitely presentable groups are approximations to groups that have no finite presentation.
Feb
27
comment Are these endofunctor categories compactly accessible? (Given a suitable base…)
Thanks again for your help here. I am going to revisit my definition for the functor, probably taking into account the tensor product...back to the drawing board
Feb
25
comment Are these endofunctor categories compactly accessible? (Given a suitable base…)
In that subcategory of CHU I linked to, we have a monoidal product. So, I am interested in functors with the following property. Given a compound system $A \otimes B$ the functor maps morphisms this way $F : f \otimes I \to I \otimes g$...I think. I haven't figured that out yet.
Feb
25
comment Examples of colimits in a category of categories
Thanks Zhen, but isn't that just in the category of sets? In the question, I mean in a category of categories.
Feb
25
awarded  Commentator
Feb
25
comment Are these endofunctor categories compactly accessible? (Given a suitable base…)
Thank you for your help here. For the base category, I am thinking of the subcategory of CHU as defined here in section 3.4. Do we still end up in the situation where all functors of the above defined type are just the identity?
Feb
25
asked Examples of colimits in a category of categories