Completeness and cocompleteness of the Kleisli category

If (T,η,μ) is a monad over a category C, which is complete and cocomplete, then what about the Kleisli category? And also, if C is cartesian closed, what about the Kleisli one?

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In general, the Kleisli category will be neither complete nor cocomplete nor cartesian closed, even if $C$ is.

You can think of the Kleisli category as the full subcategory of algebras whose objects are the free algebras. So take for example $T$ to be the monad coming from the free-forgetful adjunction between abelian groups and sets. In order for $\mathbb{Z} \times -: FreeAb \to FreeAb$ to have a right adjoint (as required by cartesian closure), it would have to preserve all colimits, for example coproducts. But $\mathbb{Z} \times (\mathbb{Z} \oplus \mathbb{Z})$ is not isomorphic to $(\mathbb{Z} \times \mathbb{Z}) \oplus (\mathbb{Z} \times \mathbb{Z})$.

Now let's tackle completeness. We can use the same example; the basic idea is that free abelian groups are not closed under (infinite) products. (In fact, a famous but nontrivial result, due to Kurosh I believe, is that the countably infinite power of $\mathbb{Z}$ is not free abelian.) But to apply this idea, one should first check that any limit in $Kl(T)$ really is constructed just as it would be in $Alg(T)$ or in $C$. That's not hard: first note that the underlying functor $U: Kl(T) \to C$ is representable (in fact, $U \cong \hom(F(1), -)$), so $U$ preserves any limits that exist in $Kl(T)$. Also, $U$ reflects isomorphisms (i.e., if $U(g)$ is an isomorphism in $C$, then $g$ is an isomorphism in $Kl(T)$). It follows that $U$ both preserves and reflects limits, and so any limit in $Kl(T)$ is constructed as it would be in $Alg(T)$ or $C$ (here, $Set$).

Finally, cocompleteness. This time I don't think the example above works quite as easily, but one example that does work is to consider the Kleisli category for the free-forgetful adjunction between $\mathbb{Z}_6$-modules and sets. I claim that the coequalizer of the pair $id, \mu_3: \mathbb{Z}_6 \to \mathbb{Z}_6$ ($\mu_3$ is multiplication by 3) does not exist in the category of free $\mathbb{Z}_6$-modules. I chose this particular coequalizer because it is an example of "splitting an idempotent"; here the idempotent is $\mu_3$. The virtue of splittings of idempotents is that they are preserved by any functor whatsoever, and reflected by any functor that reflects isomorphisms. Given this fact, it follows that splittings of idempotents in $Kl(T)$, if they exist, are constructed just as they would be in $Alg(T)$ or in $C$. But the idempotent splitting of this pair of maps in $\mathbb{Z}_6$-Mod is $\mathbb{Z}_2$, which is not free. It follows that the coequalizer doesn't exist in the Kleisli category.

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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? – Jenny Sep 9 '10 at 7:29
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. – Todd Trimble Sep 9 '10 at 8:26
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? – Todd Trimble Sep 9 '10 at 8:47
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? – Jenny Sep 9 '10 at 13:48
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! – Todd Trimble Sep 9 '10 at 20:47