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 explicitely 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.

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.

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 arguement 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 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 explicitely 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 I would to know if it appears in the litterature somewhere, or if some other result of this kind where 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.

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 explicitely 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.