Timeline for What are compact objects in the category of topological spaces?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Dec 19, 2017 at 0:37 | comment | added | R. van Dobben de Bruyn | @MikeShulman: ah, you're absolutely right. I was thinking of this definition, but this is characterising a different property. | |
| Dec 18, 2017 at 23:19 | comment | added | Mike Shulman | @R.vanDobbendeBruyn In almost all cases I'm aware of, the abstract meaning coincides with the concrete meaning. In derived categories in the homotopy-category sense (e.g. triangulated categories) directed colimits don't generally exist, so you're talking about a different notion anyway. Are you saying there is a category that has real honest directed colimits and a concrete notion of "finitely presentable" that doesn't coincide with this abstract one? | |
| Dec 18, 2017 at 15:16 | comment | added | R. van Dobben de Bruyn | @MikeShulman: for people working with derived categories (say of $A$-modules), the word finitely presentable is not great because it already has a concrete meaning for $A$-modules. This might be why different terminology was chosen, although admittedly I don't know the history. | |
| Dec 18, 2017 at 8:39 | comment | added | Philippe Gaucher | The presentable objects in the category of general topological spaces are the discrete ones (it's a very known joke). A topological space is $\lambda$-presentable if and only if it is discrete of cardinal less than $\lambda$. Your question is the case $\lambda=\aleph_0$. | |
| Dec 18, 2017 at 6:47 | comment | added | Mike Shulman | This is why I've always disliked the use of the word "compact" for the property in your first paragraph. An alternative, which also has a longer weight of tradition behind it, is "finitely presentable". | |
| Dec 16, 2017 at 23:22 | history | edited | R. van Dobben de Bruyn | CC BY-SA 3.0 |
Wrong implication.
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| S Dec 16, 2017 at 23:13 | answer | added | R. van Dobben de Bruyn | timeline score: 12 | |
| S Dec 16, 2017 at 23:13 | history | asked | R. van Dobben de Bruyn | CC BY-SA 3.0 |