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A category-theoretic angle. Accordingly, more questions than answers. But at least I hope it uncovers some subtle issue.

Let us ask how to define transfinite iterates of an endofunctor $F$. That is, what properties should uniquely (up to isomorphism) determine $F^\lambda$ as something like "$\underbrace{\cdots\circ F\circ F\circ\cdots}_{\text{$\lambda$ times}}$".

If this is solved, then $\kappa\uparrow\lambda$ could be defined as the cardinality of $F^\lambda(\text{singleton})$, where $F$ is the functor from sets to sets given by $F(S)=X^S$, with $X$ any (fixed) set of cardinality $\kappa$. (It is true that $F$ is actually contravariant, but there still is no problem iterating it.)

When $\lambda$ is an ordinal, one might try direct limit of the $\lambda$-shaped diagram, with $F$ as transition functors.

In this way one gets something peculiar. Already at $\lambda=\omega$ the limiting category might be not equivalent to the category of sets. Whatever it is, $F^\lambda(\text{singleton})$ will be defined, it will be an isomorphism class of objects of a category defined uniquely up to equivalence.

So it remains to describe this category, at least the class of isomorphism classes of its objects. This will then give a natural candidate for $\kappa\uparrow\lambda$ as one of the members of this class, uniquely determined by the above.

A variant - we could take the (large) groupoid of sets and bijections, we don't actually need non-bijective morphisms. However I don't know whether we would obtain something essentially different at limiting stages.

Still another variant - we might take the $\textit{inverse}$ limit of the backwards diagram; if it does have a terminal object, we might take the value of the limiting functor on it. This then would be a genuine set. Here however I don't know whether the terminal indeed exists in the limiting category.

When $\lambda$ is just a cardinal, I am at a loss. I need some ordering to get started with the above.

After a while I decided to post a separate question about the nature of these limiting categories, Infinite iterates of the contravariant hom endofunctors on sets

A category-theoretic angle. Accordingly, more questions than answers. But at least I hope it uncovers some subtle issue.

Let us ask how to define transfinite iterates of an endofunctor $F$. That is, what properties should uniquely (up to isomorphism) determine $F^\lambda$ as something like "$\underbrace{\cdots\circ F\circ F\circ\cdots}_{\text{$\lambda$ times}}$".

If this is solved, then $\kappa\uparrow\lambda$ could be defined as the cardinality of $F^\lambda(\text{singleton})$, where $F$ is the functor from sets to sets given by $F(S)=X^S$, with $X$ any (fixed) set of cardinality $\kappa$. (It is true that $F$ is actually contravariant, but there still is no problem iterating it.)

When $\lambda$ is an ordinal, one might try direct limit of the $\lambda$-shaped diagram, with $F$ as transition functors.

In this way one gets something peculiar. Already at $\lambda=\omega$ the limiting category might be not equivalent to the category of sets. Whatever it is, $F^\lambda(\text{singleton})$ will be defined, it will be an isomorphism class of objects of a category defined uniquely up to equivalence.

So it remains to describe this category, at least the class of isomorphism classes of its objects. This will then give a natural candidate for $\kappa\uparrow\lambda$ as one of the members of this class, uniquely determined by the above.

A variant - we could take the (large) groupoid of sets and bijections, we don't actually need non-bijective morphisms. However I don't know whether we would obtain something essentially different at limiting stages.

Still another variant - we might take the $\textit{inverse}$ limit of the backwards diagram; if it does have a terminal object, we might take the value of the limiting functor on it. This then would be a genuine set. Here however I don't know whether the terminal indeed exists in the limiting category.

When $\lambda$ is just a cardinal, I am at a loss. I need some ordering to get started with the above.

After a while I decided to post a separate question about the nature of these limiting categories, Infinite iterates of the contravariant hom endofunctors on sets

A category-theoretic angle. Accordingly, more questions than answers. But at least I hope it uncovers some subtle issue.

Let us ask how to define transfinite iterates of an endofunctor $F$. That is, what properties should uniquely (up to isomorphism) determine $F^\lambda$ as something like "$\underbrace{\cdots\circ F\circ F\circ\cdots}_{\text{$\lambda$ times}}$".

If this is solved, then $\kappa\uparrow\lambda$ could be defined as the cardinality of $F^\lambda(\text{singleton})$, where $F$ is the functor from sets to sets given by $F(S)=X^S$, with $X$ any (fixed) set of cardinality $\kappa$. (It is true that $F$ is actually contravariant, but there still is no problem iterating it.)

When $\lambda$ is an ordinal, one might try direct limit of the $\lambda$-shaped diagram, with $F$ as transition functors.

In this way one gets something peculiar. Already at $\lambda=\omega$ the limiting category might be not equivalent to the category of sets. Whatever it is, $F^\lambda(\text{singleton})$ will be defined, it will be an isomorphism class of objects of a category defined uniquely up to equivalence.

So it remains to describe this category, at least the class of isomorphism classes of its objects. This will then give a natural candidate for $\kappa\uparrow\lambda$ as one of the members of this class, uniquely determined by the above.

A variant - we could take the (large) groupoid of sets and bijections, we don't actually need non-bijective morphisms. However I don't know whether we would obtain something essentially different at limiting stages.

Still another variant - we might take the $\textit{inverse}$ limit of the backwards diagram; if it does have a terminal object, we might take the value of the limiting functor on it. This then would be a genuine set. Here however I don't know whether the terminal indeed exists in the limiting category.

When $\lambda$ is just a cardinal, I am at a loss. I need some ordering to get started with the above.

After a while I decided to post a separate question about the nature of these limiting categories, Infinite iterates of the contravariant hom endofunctors on sets

On the afterthought -

One probably might even squeeze this back to an endofunctor of the category $\mathbf{Set}$ of sets: there is a cone from the diagram considered to $\mathbf{Set}$, by composing with $F$ at successor ordinals and taking Kan extensions at limit ordinals. This will induce a functor from the colimit back to $\mathbf{Set}$. Dually for the inverse limit case.

A category-theoretic angle. Accordingly, more questions than answers. But at least I hope it uncovers some subtle issue.

Let us ask how to define transfinite iterates of an endofunctor $F$. That is, what properties should uniquely (up to isomorphism) determine $F^\lambda$ as something like "$\underbrace{\cdots\circ F\circ F\circ\cdots}_{\text{$\lambda$ times}}$".

If this is solved, then $\kappa\uparrow\lambda$ could be defined as the cardinality of $F^\lambda(\text{singleton})$, where $F$ is the functor from sets to sets given by $F(S)=X^S$, with $X$ any (fixed) set of cardinality $\kappa$. (It is true that $F$ is actually contravariant, but there still is no problem iterating it.)

When $\lambda$ is an ordinal, one might try direct limit of the $\lambda$-shaped diagram, with $F$ as transition functors.

In this way one gets something peculiar. Already at $\lambda=\omega$ the limiting category might be not equivalent to the category of sets. Whatever it is, $F^\lambda(\text{singleton})$ will be defined, it will be an isomorphism class of objects of a category defined uniquely up to equivalence.

So it remains to describe this category, at least the class of isomorphism classes of its objects. This will then give a natural candidate for $\kappa\uparrow\lambda$ as one of the members of this class, uniquely determined by the above.

A variant - we could take the (large) groupoid of sets and bijections, we don't actually need non-bijective morphisms. However I don't know whether we would obtain something essentially different at limiting stages.

Still another variant - we might take the $\textit{inverse}$ limit of the backwards diagram; if it does have a terminal object, we might take the value of the limiting functor on it. This then would be a genuine set. Here however I don't know whether the terminal indeed exists in the limiting category.

When $\lambda$ is just a cardinal, I am at a loss. I need some ordering to get started with the above.

After a while I decided to post a separate question about the nature of these limiting categories, Infinite iterates of the contravariant hom endofunctors on sets

A category-theoretic angle. Accordingly, more questions than answers. But at least I hope it uncovers some subtle issue.

Let us ask how to define transfinite iterates of an endofunctor $F$. That is, what properties should uniquely (up to isomorphism) determine $F^\lambda$ as something like "$\underbrace{\cdots\circ F\circ F\circ\cdots}_{\text{$\lambda$ times}}$".

If this is solved, then $\kappa\uparrow\lambda$ could be defined as the cardinality of $F^\lambda(\text{singleton})$, where $F$ is the functor from sets to sets given by $F(S)=X^S$, with $X$ any (fixed) set of cardinality $\kappa$. (It is true that $F$ is actually contravariant, but there still is no problem iterating it.)

When $\lambda$ is an ordinal, one might try direct limit of the $\lambda$-shaped diagram, with $F$ as transition functors.

In this way one gets something peculiar. Already at $\lambda=\omega$ the limiting category might be not equivalent to the category of sets. Whatever it is, $F^\lambda(\text{singleton})$ will be defined, it will be an isomorphism class of objects of a category defined uniquely up to equivalence.

So it remains to describe this category, at least the class of isomorphism classes of its objects. This will then give a natural candidate for $\kappa\uparrow\lambda$ as one of the members of this class, uniquely determined by the above.

A variant - we could take the (large) groupoid of sets and bijections, we don't actually need non-bijective morphisms. However I don't know whether we would obtain something essentially different at limiting stages.

Still another variant - we might take the $\textit{inverse}$ limit of the backwards diagram; if it does have a terminal object, we might take the value of the limiting functor on it. This then would be a genuine set. Here however I don't know whether the terminal indeed exists in the limiting category.

When $\lambda$ is just a cardinal, I am at a loss. I need some ordering to get started with the above.

After a while I decided to post a separate question about the nature of these limiting categories, Infinite iterates of the contravariant hom endofunctors on sets

A category-theoretic angle. Accordingly, more questions than answers. But at least I hope it uncovers some subtle issue.

Let us ask how to define transfinite iterates of an endofunctor $F$. That is, what properties should uniquely (up to isomorphism) determine $F^\lambda$ as something like "$\underbrace{\cdots\circ F\circ F\circ\cdots}_{\text{$\lambda$ times}}$".

If this is solved, then $\kappa\uparrow\lambda$ could be defined as the cardinality of $F^\lambda(\text{singleton})$, where $F$ is the functor from sets to sets given by $F(S)=X^S$, with $X$ any (fixed) set of cardinality $\kappa$. (It is true that $F$ is actually contravariant, but there still is no problem iterating it.)

When $\lambda$ is an ordinal, one might try direct limit of the $\lambda$-shaped diagram, with $F$ as transition functors.

In this way one gets something peculiar. Already at $\lambda=\omega$ the limiting category might be not equivalent to the category of sets. Whatever it is, $F^\lambda(\text{singleton})$ will be defined, it will be an isomorphism class of objects of a category defined uniquely up to equivalence.

So it remains to describe this category, at least the class of isomorphism classes of its objects. This will then give a natural candidate for $\kappa\uparrow\lambda$ as one of the members of this class, uniquely determined by the above.

A variant - we could take the (large) groupoid of sets and bijections, we don't actually need non-bijective morphisms. However I don't know whether we would obtain something essentially different at limiting stages.

Still another variant - we might take the $\textit{inverse}$ limit of the backwards diagram; if it does have a terminal object, we might take the value of the limiting functor on it. This then would be a genuine set. Here however I don't know whether the terminal indeed exists in the limiting category.

When $\lambda$ is just a cardinal, I am at a loss. I need some ordering to get started with the above.

After a while I decided to post a separate question about the nature of these limiting categories, Infinite iterates of the contravariant hom endofunctors on sets

On the afterthought -

One probably might even squeeze this back to an endofunctor of the category $\mathbf{Set}$ of sets: there is a cone from the diagram considered to $\mathbf{Set}$, by composing with $F$ at successor ordinals and taking Kan extensions at limit ordinals. This will induce a functor from the colimit back to $\mathbf{Set}$. Dually for the inverse limit case.