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The following theorem is relatively classical:

Theorem: Given an accessible endofunctor, (co)pointed endofunctor or (co)monad $T$ on a locally presentable category $C$, then the category of $T$-(co)algebra is also locally presentable.

The proof goes as follows: in each case the category of (co)algebra can be written as a certain weighted bilimits in the category of accessible categories and accessible functors hence it is an accessible category. Moreover it is well know that categories of algebras are complete and categories of co-algebras are co-complete (in both case either limits or colimits are created by the forgetfull functor) so in both case they are locally presentable categories.

Unfortunately the argument above give very little control on the presentability rank of the category of (co)algebras. And this what this question is about: can we give a good bound on the presentability rank of the categories of (co)algebras ?

In the special case of algebra on a monad it is easy to see explicitly that if $C$ is locally $\lambda$-presentable and $T$ is $\lambda$-accessible then the category of $T$ algebras is locally $\lambda$-presentable, by showing that the free algebra on $\kappa$-presentable objects form a dense subcategory of $\kappa$-presentable objects. This done for example in Bird's Phd thesis (and probably in other places as well).

I convinced myself that the following was true:

Conjecture: Given $\kappa$ an uncountable regular cardinal. If in the theorem above $C$ is locally $\kappa$-presentable and $T$ is $\kappa$-accessible then the category of $T$-(co)-algebras is locally $\kappa$-presentable.

Assuming it is correct, I like would to know if it was proved somewhere, or if some other result of this kind is known (or if on the contrary counter-example where known) or not.

I'm stating both the case of algebras and coalgebras, but I am considerably more interested by the case of coalgebras.

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  • $\begingroup$ Do you have a good example of a failure of this when $\kappa =\omega$? $\endgroup$ – Saal Hardali Jan 26 at 16:01
  • $\begingroup$ @SaalHardali : I would be interested to have one ! What I have so far is the following: If you look at the identity endofunctor I on the category of set, the category of I-coalgebra is $\omega$-presentable, but the forgetfull functor do not send finitely presentable object to finitely presentable object (which is always true and essential part of the proof when $\kappa > \omega$). I would be interested to see an example with a co-monad and one where the category of coalgebra fails to be $\omega$-presentable... $\endgroup$ – Simon Henry Jan 26 at 16:13
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    $\begingroup$ That's a good counterexample for the case of coalgebras over just endofunctors. But I must say I have little to no experience with this notion so I can't decide if it counts as positive or negative evidence towards the statement for Comonads... Which I would be delighted to have a proof/counterexample for. $\endgroup$ – Saal Hardali Jan 26 at 20:04
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    $\begingroup$ I think I have an example for failure when $\kappa = \omega$. If we take $C = \otimes^{\infty}_{k=0} E\{x_k\}$ a countable tensor product of simple coalgebras cogenerated by a single primitive element (over $\mathbb{F}_2$). Then in the $\infty$-category of $C$-comodules (in $\mathbb{F}_2$-complexes) there are no non-trivial compact objects. Essentially because for any compact comodule there's an non-nilpotent element in the Ext of the identity functor which acts on it's underlying complex by 0. $\endgroup$ – Saal Hardali Jan 30 at 19:08
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    $\begingroup$ @LeonidPositselski : I don't know one that does all the cases, but I think for each special case I can give you a reference. It follows from the existence of bilimits in the category of accessible category and accessible functor between them (as explained in the paragraph just below the theorem) and this results can be for example found as theorem 5.1.6 in Makkai & Parré "Accessible categories", or a bit in pieces in Rosicky-Adamek locally presentable catgories (they treat lax limits in section 2.H and other case as exercice 2.n and 2.m, which putting everything together give all bilimits. $\endgroup$ – Simon Henry Mar 1 at 18:10
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The case of algebras for a monad is discussed explicitly in Gregory Bird's thesis (see theorem 6.9). The case of the categories of algebras for an endofunctor or pointed endofunctor can be deduced from the fact that if $F$ is a (pointed) endofunctor on $C$, then $F$-Alg $\rightarrow C$ obviously satisfies the condition of Beck's monadicity theorem, and the induced monad preserves $\lambda$-filtered colimits if $F$ does. All this works for any regular $\lambda$, even $\omega$.

For the case of coalgebras, Jiří Rosický pointed out the key references to me by email:

The following theorem is due to Adámek and Porst in On tree coalgebras and coalgebra presentations as their Theorem 4.2.

We fix $\lambda$ an uncountable regular cardinal.

Theorem: Let $A$ be a $\lambda$-accessible category that admits colimits of $\omega$-chains, and let $F: A \rightarrow A$ be a $\lambda$-accessible endofunctor. Then:

  • The category of $F$-coalgebra is $\lambda$-accessible.
  • A $F$-coalgebra is $\lambda$-presentable if and only if its underlying object is $\lambda$-presentable in $A$.

Corollary: If $A$ is a locally $\lambda$-presentable category and $F$ is a $\lambda$-accessible endofunctor on $A$ then the category of $F$-coalgebra is locally $\lambda$-presentable.

The corollary follows immediately: as $A$ is cocomplete it has colimits of $\omega$-chains, and the forgetful functor $F$-coalg $\rightarrow A$ create colimits, so $F$-coalg is $\lambda$-accessible and cocomplete, hence $\lambda$-presentable.

We can immediately deduce that:

Theorem: If $F$ is a $\lambda$-accessible copointed endofunctor or comonad on a locally $\lambda$-presentable $A$, then:

  • The category of $F$-coalgebras is locally $\lambda$-presentable.
  • An $F$-coalgebra is $\lambda$-presentable if and only if its underlying object is $\lambda$-presentable.

Indeed, this can be deduced from the corollary above using that (for $\lambda$ an uncountable cardinal) the category of $\lambda$-presentable categories and left adjoint functors between them preserving $\lambda$-presentable objects is closed under $\lambda$-small cat weighted pseudo-limits. The category of $M$-coalgebras for a copointed endofunctor $M$ can be constructed as a full subcategory of the category of $M_0$-coalgebra where $M_0$ is the underlying endofunctor of $M$ as the equifier of $Id,v:U \rightrightarrows U$ where $U:M_0\text{-Coalg} \rightarrow C$ is the forgetful functor, and $v$ is the natural transformation which on each $M_0$-coalgebra $X$ is the composite $X \rightarrow M(X) \rightarrow X$.

When $M$ is a comonad this is a bit more complicated as we would like to take the equifier of the two natural transformation $X \rightrightarrows M_0^2(X)= M_0(M_0(X))$ corresponding to the two side of the usual square, but as $F^S$ is not a left adjoint functor we cannot directly conclude using 2-limits of diagrams of left adjoint functors.

Instead we consider the category: $$E=\{X \in C, v_1,v_2:X \rightrightarrows M_0^2(X) \}$$

which is the category of coalgebra for the endofunctor: $$ X \mapsto M_0^2(X) \times M_O(X)^2$$

which is indeed $\lambda$-accessible, so $E$ is locally $\lambda$-presentable and its $\lambda$-presentable objects are these whose underlying object $X$ is $\lambda$-presentable.

One has a natural functor $M_0$-coalg to $E$ which sends each $M_0$-algebra to the pair of maps $X \rightrightarrows M_0^2 $ corresponding to square defining $M$-algebras and another functor from $M_0^2$-Coalg to $E$ that sends each $f:X \rightarrow M^2_0(X)$ to $(X,f,f)$. taking the (pseudo)pullback of these two functors give us exactly the category of $M_0$-coalgebras compatible with the comultiplication of $M$. Both these functors clearly preserve all colimits and $\lambda$-presentable objects, so by the results mentioned above, this category is locally $\lambda$-presentable. Combining this with the case of copointed endofunctors we obtain the result.

I've included this material with a bit more details and other related results in appendix A of this paper.


Regarding relaxing the assumption that $\lambda$ is uncountable, Adámek and Porst show in their paper that the endofunctor:

$$ \mathcal{P}_f(X) = \{ F \subset X | F \text{ is finite} \} $$

as an endofunctor of the category of sets (with the direct image functoriality) is a counter example to the first theorem in the case $\lambda=\omega$. That is the category of $\mathcal{P}_f$ coalgebra is not finitely accessible. For the case of comonads, there seems to be a counter-example in the comments of the question.

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    $\begingroup$ This is super cool! Why didn't I know of it?! $\endgroup$ – Ivan Di Liberti Feb 24 at 22:30

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