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The following point is of course related to the fact that you can use models different from Set, but I think it deserves to be discussed explicitly.

Every clone (in the unverisal algebra sense) can of course be written as a set of morphisms in a suitable category. This category can always be chosen to be the category of sets (since the functions in a clone are set-functions), but it might also very well be another category. For instance, one can write every clone on a finite set as the set of homomorphisms over a relational structure in a variety of relational structures (understood as a category). This works since every clone on a finite set is the set of polymorphisms of a certain set of relations.

To give another example: If the clone in question is a so-called centralizer clone, then it can even be written as the set of homomorphisms in a variety of algebras (understood as a category).

Thus, in terms of Lawvere theories, you look at models in categories different from Set. The advantage of the category-theoretic setting is that you can now apply duality theory. For instance, if we take the centralizer clone of a Boolean algebra, then we can interpret it as a model in the category of Boolean algebras and use the Stone duality to dualize it to a comodel in the category of sets, which are then, if you transfer the scenario back to unviversal algebra, essentially the (well-studied and easy to understand) coclones over sets.

In the last few years, there have been a few papers that studied clones in exactly this way. In particular, this gave many new results for the centralizer clones of Boolean algebras, distributive lattices (which, via Priestley Duality, become comodels in the category of Priestley spaces) and median algebras (which, via a duality by Isbell, become comodels in the category of what I think is often called Isbell spaces). I do not see any (convenient) way to formulate such a connection without using category theory and looking at clones as Lawvere theories.

I believe this is a nice example where shifting to a category theoretic setting has some actual advantages and is not just a question of whether you personally like it or not.

The following point is of course related to the fact that you can use models different from Set, but I think it deserves to be discussed explicitly.

Every clone (in the unverisal algebra sense) can be written as a set of morphisms in a suitable category. Obviously, this category can be chosen to be the category of sets (since the functions in a clone are set-functions), but we might also very well chose another category. For instance, one can write every clone on a finite set as the set of homomorphisms over a relational structure in a variety of relational structures (understood as a category). This works since every clone on a finite set is the set of polymorphisms of a certain set of relations.

To give another example: If the clone in question is a so-called centralizer clone, then it can even be written as the set of homomorphisms in a variety of algebras (understood as a category).

Thus, in terms of Lawvere theories, you look at models in categories different from Set. The advantage of the category-theoretic setting is that you can now apply duality theory. For instance, if we take the centralizer clone of a Boolean algebra, then we can interpret it as a model in the category of Boolean algebras and use the Stone duality to dualize it to a comodel in the category of sets, which are then, if you transfer the scenario back to unviversal algebra, essentially the (well-studied and easy to understand) coclones over sets.

In the last few years, there have been a few papers that studied clones in exactly this way. In particular, this gave many new results for the centralizer clones of Boolean algebras, distributive lattices (which, via Priestley Duality, become comodels in the category of Priestley spaces) and median algebras (which, via a duality by Isbell, become comodels in the category of what I think is often called Isbell spaces). I do not see any (convenient) way to formulate such a connection without using category theory and looking at clones as Lawvere theories.

I believe this is a nice example where shifting to a category theoretic setting has some actual advantages and is not just a question of whether you personally like it or not.

The following point is of course related to the fact that you can use models different from Set, but I think it deserves to be discussed explicitly.

Every clone (in the unverisal algebra sense) can of course be written as a set of morphisms in a suitable category. This category can always be chosen to be the category of sets (since the functions in a clone are set-functions), but it might also very well be another category. For instance, one can write every clone on a finite set as the set of homomorphisms over a relational structure in a variety of relational structures (understood as a category). This works since every clone on a finite set is the set of polymorphisms of a certain set of relations.

To give another example: If the clone in question is a so-called centralizer clone, then it can even be written as the set of homomorphisms in a variety of algebras (understood as a category).

Thus, in terms of Lawvere theories, you look at models in categories different from Set. The advantage of the category-theoretic setting is that you can now apply duality theory. For instance, if we take the centralizer clone of a Boolean algebra, then we can interpret it as a model in the category of Boolean algebras and use the Stone duality to dualize it to a comodel in the category of sets, which are then, if you transfer the scenario back to unviversal algebra, essentially the (well-studied and easy to understand) coclones over sets.

In the last few years, there have been a few papers that studied clones in exactly this way. In particular, this gave many new results for clones the centralizer clones of Boolean algebras, distributive lattices (which, via Priestley Duality, become comodels in the category of Priestley spaces) and median algebras (which, via a duality by Isbell, become comodels in the category of what I think is often called Isbell spaces). I do not see any (convenient) way to formulate such a connection without using category theory and looking at clones as Lawvere theories.

The following point is of course related to the fact that you can use models different from Set, but I think it deserves to be discussed explicitly.

Every clone (in the unverisal algebra sense) can of course be written as a set of morphisms in a suitable category. This category can always be chosen to be the category of sets (since the functions in a clone are set-functions), but it might also very well be another category. For instance, one can write every clone on a finite set as the set of homomorphisms over a relational structure in a variety of relational structures (understood as a category). This works since every clone on a finite set is the set of polymorphisms of a certain set of relations.

To give another example: If the clone in question is a so-called centralizer clone, then it can even be written as the set of homomorphisms in a variety of algebras (understood as a category).

Thus, in terms of Lawvere theories, you look at models in categories different from Set. The advantage of the category-theoretic setting is that you can now apply duality theory. For instance, if we take the centralizer clone of a Boolean algebra, then we can interpret it as a model in the category of Boolean algebras and use the Stone duality to dualize it to a comodel in the category of sets, which are then, if you transfer the scenario back to unviversal algebra, essentially the (well-studied and easy to understand) coclones over sets.

In the last few years, there have been a few papers that studied clones in exactly this way. In particular, this gave many new results for the centralizer clones of Boolean algebras, distributive lattices (which, via Priestley Duality, become comodels in the category of Priestley spaces) and median algebras (which, via a duality by Isbell, become comodels in the category of what I think is often called Isbell spaces). I do not see any (convenient) way to formulate such a connection without using category theory and looking at clones as Lawvere theories.

I believe this is a nice example where shifting to a category theoretic setting has some actual advantages and is not just a question of whether you personally like it or not.

The following point is of course related to the fact that you can use models different from Set, but I think it deserves to be discussed explicitly.

Every clone (in the unverisal algebra sense) can of course be written as a set of morphisms in a suitable category. This category can always be chosen to be the category of sets (since the functions in a clone are set-functions), but it might also very well be another category. For instance, one can write every clone on a finite set as the set of homomorphisms over a relational structure in a variety of relational structures (understood as a category). This works since every clone on a finite set is the set of polymorphisms of a certain set of relations.

To give another example: If the clone in question is a so-called centralizer clone, then it can even be written as the set of homomorphisms in a variety of algebras (understood as a category).

Thus, in terms of Lawvere theories, you look at models in categories different from Set. The advantage of the category-theoretic setting is that you can now apply duality theory. For instance, if we take the centralizer clone of a Boolean algebra, then we can interpret it as a model in the category of Boolean algebras and use the Stone duality to dualize it to a comodel in the category of sets, which are then, if you transfer the scenario back to unviversal algebra, essentially the (well-studied and easy to understand) coclones over sets.

In the last few years, there have been a few papers that studied clones in exactly this way. In particular, this gave many new results for clones the centralizer clones of Boolean algebras, distributive lattices and median algebras. I do not see any (convenient) way to formulate such a connection without using category theory and looking at clones as Lawvere theories.

The following point is of course related to the fact that you can use models different from Set, but I think it deserves to be discussed explicitly.

Every clone (in the unverisal algebra sense) can of course be written as a set of morphisms in a suitable category. This category can always be chosen to be the category of sets (since the functions in a clone are set-functions), but it might also very well be another category. For instance, one can write every clone on a finite set as the set of homomorphisms over a relational structure in a variety of relational structures (understood as a category). This works since every clone on a finite set is the set of polymorphisms of a certain set of relations.

To give another example: If the clone in question is a so-called centralizer clone, then it can even be written as the set of homomorphisms in a variety of algebras (understood as a category).

Thus, in terms of Lawvere theories, you look at models in categories different from Set. The advantage of the category-theoretic setting is that you can now apply duality theory. For instance, if we take the centralizer clone of a Boolean algebra, then we can interpret it as a model in the category of Boolean algebras and use the Stone duality to dualize it to a comodel in the category of sets, which are then, if you transfer the scenario back to unviversal algebra, essentially the (well-studied and easy to understand) coclones over sets.

In the last few years, there have been a few papers that studied clones in exactly this way. In particular, this gave many new results for clones the centralizer clones of Boolean algebras, distributive lattices (which, via Priestley Duality, become comodels in the category of Priestley spaces) and median algebras (which, via a duality by Isbell, become comodels in the category of what I think is often called Isbell spaces). I do not see any (convenient) way to formulate such a connection without using category theory and looking at clones as Lawvere theories.