Timeline for Pushouts in the category of adjunctions
Current License: CC BY-SA 3.0
11 events
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| Mar 19, 2014 at 20:26 | history | edited | Will Sawin | CC BY-SA 3.0 |
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| S Feb 17, 2014 at 16:23 | history | suggested | Wolfgang Jeltsch | CC BY-SA 3.0 |
improved formatting
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| Feb 17, 2014 at 16:20 | review | Suggested edits | |||
| S Feb 17, 2014 at 16:23 | |||||
| Jan 25, 2014 at 20:19 | history | edited | Will Sawin | CC BY-SA 3.0 |
added 873 characters in body
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| Jan 25, 2014 at 20:07 | history | edited | Will Sawin | CC BY-SA 3.0 |
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| Jan 25, 2014 at 20:04 | comment | added | Will Sawin | Good point. What are good examples of categories without filtered colimits? I don't know any way to find them other than enumerating all the categories I know and checking if they have filtered colimits, which seems tiresome. | |
| Jan 25, 2014 at 17:41 | comment | added | Daniel Schäppi | On the other hand, if you consider the 2-category of say lfp categories and adjunctions whose right adjoint does preserve filtered colimits, then you do get the desired pushouts. This 2-category is equivalent to finitely cocomplete categories with functors preserving finite colimits, and this is 2-monadic over the category of small categories. Since the 2-monad in question has rank, this 2-category has all bicolimits. I think it would be interesting to see if there are counterexamples if the categories involved do not have (many) colimits. | |
| Jan 25, 2014 at 17:35 | comment | added | Daniel Schäppi | I don't think this works quite as intended. First of all, you often talk about limits when you really mean colimits (both in the definition of $X_i$ and $X_{\infty}$). But right adjoints need not preserve colimits (not even filtered colimits in general), so your argument that $G_1 X_{\infty}=G_2 Y_{\infty}$ does not work. | |
| Jan 25, 2014 at 16:58 | history | edited | Will Sawin | CC BY-SA 3.0 |
edited body
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| Jan 25, 2014 at 10:15 | comment | added | Oskar | Small typo: "Then by the same logic we have a natural map $Hom_{\mathcal D_1}(X_0,X) \to Hom_{\mathcal D_1} ( F_2 G_1 Y_1, X)$" - $F_1 G_2 Y_1$ instead of $F_2 G_1 Y_1$. | |
| Jan 24, 2014 at 22:53 | history | answered | Will Sawin | CC BY-SA 3.0 |