Timeline for Completeness and cocompleteness of the Kleisli category
Current License: CC BY-SA 2.5
7 events
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| Sep 9, 2010 at 20:47 | comment | added | Todd Trimble | I don't know, Jenny. It might be worthwhile posting this as a separate question at MO (preferably with a link to a suitable document with relevant definitions). Or, perhaps you know an expert in domain theory you can ask directly? Good luck! | |
| Sep 9, 2010 at 13:48 | comment | added | Jenny | Take the Probabilistic Powerdomain of Evaluations as example, the base category is Dcpo, directed complete posets with Scott continuous functions, does the finite product of the Kleisli category of this monad exist? | |
| Sep 9, 2010 at 8:47 | comment | added | Todd Trimble | There are many weird examples where this can happen. A silly example is where you take a non-distributive lattice $L$ and define $T: L \to L$ to take the bottom element 0 to 0, and every other element to the top element 1. The Eilenberg-Moore category is then the two-element lattice which is cartesian closed. Slightly less silly: take $C = Top/X$, which is generally not cartesian closed, and $T$ to take a map $f: Y \to X$ to its image $im(f) \hookrightarrow X$. The EM category is then the lattice of subspaces of $X$ which is cartesian closed. But why do you ask? | |
| Sep 9, 2010 at 8:26 | comment | added | Todd Trimble | It's hard to give reasonable general answers. I can say that if $C$ has coproducts, so does $Kl(T)$. Depending on $C$ and the monad $T$, it may be that $Kl(T)$ is complete and cocomplete; for example if $C$ is bicomplete and every $T$-algebra is free. This happens e.g. when $T$ is an idempotent monad; details at ncatlab.org/nlab/show/completion. But usually it comes down to individual cases and asking questions like: are products of free objects free? Are subobjects of free objects free? If you do have a specific example, maybe we can discuss that. Ans. to 2nd question next comment. | |
| Sep 9, 2010 at 7:29 | comment | added | Jenny | Thank you. But I am not clear that when and where the limit or colimit of the Kleisli category exists? Another question, if the base category is not cartesian closed, is it possible that the Elienberg-Moore category cartesian closed? | |
| Sep 8, 2010 at 15:11 | history | edited | Todd Trimble | CC BY-SA 2.5 |
Corrected a small arithmetic mistake at the end.
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| Sep 7, 2010 at 19:28 | history | answered | Todd Trimble | CC BY-SA 2.5 |