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When is the opposite of the category of algebras of a Lawvere theory an extensive category? Any necessary or sufficient conditions on the Lawvere theory will be interesting to me.

Here's why I'm interested. An extensive category acts to some extent like a category of "spaces" (e.g. topological spaces, schemes, sets, etc.). We can see this in various ways. First, in any extensive category, an object over $X + Y$ is the same as an object over $X$ together with an object over $Y$. Second, if such a category has binary products, they distribute over binary coproducts:

$$ X \times (Y + Z) \cong X \times Y + X \times Z.$$

Third, in any extensive category, any object with a morphism to the initial object $0$ must be initial itself. All these are basic things we expect of a category of "spaces".

Furthermore, algebras of a Lawvere theory are a pretty good formalization of the general idea of "algebraic gadgets" (e.g. monoids, rings, $R$-algebras, etc.).

So, my question is an attempt at getting at the vaguer question "when does the opposite of a category of algebraic gadgets act like a category of spaces?"

One classic example is the Lawvere theory of commutative rings. Its category of algebras is CommRing, and the opposite of that is the category of affine schemes, which is extensive.

Another classic example is the Lawvere theory of commutative $R$-algebras for some commutative ring $R$. The opposite of its category of algebras is the category of affine schemes over $R$, which is again extensive.

But there should also be some more exotic examples, showing up e.g. in Durov's approach to algebraic geometry.

So, it would be nice to get more of a sense of the full range of examples: Lawvere theories whose category of algebras has extensive opposite.

When is the opposite of the category of algebras of a Lawvere theory an extensive category? Any necessary or sufficient conditions on the Lawvere theory will be interesting to me.

Here's why I'm interested. An extensive category acts to some extent like a category of "spaces" (e.g. topological spaces, schemes, sets, etc.). We can see this in various ways. First, in any extensive category, an object over $X + Y$ is the same as an object over $X$ together with an object over $Y$. Second, if such a category has binary products, they distribute over binary coproducts:

$$ X \times (Y + Z) \cong X \times Y + X \times Z.$$

Third, in any extensive category, any object with a morphism to the initial object $0$ must be initial itself. All these are basic things we expect of a category of "spaces".

Furthermore, algebras of a Lawvere theory are a pretty good formalization of the general idea of "algebraic gadgets" (e.g. monoids, rings, $R$-algebras, etc.).

So, my question is an attempt at getting at the vaguer question "when does the opposite of a category of algebraic gadgets act like a category of spaces?"

One classic example is the Lawvere theory of commutative rings. Its category of algebras is CommRing, and the opposite of that is the category of affine schemes, which is extensive.

Another classic example is the Lawvere theory of commutative $R$-algebras for some commutative ring $R$. The opposite of its category of algebras is the category of affine schemes over $R$, which is again extensive.

But there are also more exotic examples, some of which show up e.g. in Durov's approach to algebraic geometry. For example the Lawvere theories for commutative rigs, distributive lattices or $C^\infty$-rings also have categories of algebras with extensive opposites.

So, it would be nice to get a clearer sense of the full range of Lawvere theories whose category of algebras have extensive opposites. Are there any that do not involve a commutative rig structure?

When is the opposite of the category of algebras of a Lawvere theory an extensive category? Any necessary or sufficient conditions on the Lawvere theory will be interesting to me.

Here's why I'm interested. An extensive category acts to some extent like a category of "spaces" (e.g. topological spaces, schemes, sets, etc.). We can see this in various ways. First, in any extensive category, an object over $X + Y$ is the same as an object over $X$ together with an object over $Y$. Second, if such a category has binary products, they distribute over binary coproducts:

$$ X \times (Y + Z) \cong X \times Y + X \times Z$$

Third, in any extensive category, any object with a morphism to to the initial object $0$ must be initial itself. All these are basic things we expect of a category of "spaces".

Furthermore, algebras of a Lawvere theory are a pretty good formalization of the general idea of "algebraic gadgets" (e.g. monoids, rings, $R$-algebras, etc.).

So, my question is an attempt at getting at the vaguer question "when does the opposite of a category of algebraic gadgets act like a category of spaces?"

One classic example is the Lawvere theory of commutative rings. Its category of algebras is CommRing, and the opposite of that is the category of affine schemes, which is extensive.

Another classic example is the Lawvere theory of commutative $R$-algebras for some commutative ring $R$. The opposite of its category of algebras is the category of affine schemes over $R$, which is again extensive.

But there should also be some more exotic examples, showing up e.g. in Durov's approach to algebraic geometry.

So, it would be nice to get more of a sense of the full range of examples: Lawvere theories whose category of algebras has extensive opposite.

When is the opposite of the category of algebras of a Lawvere theory an extensive category? Any necessary or sufficient conditions on the Lawvere theory will be interesting to me.

Here's why I'm interested. An extensive category acts to some extent like a category of "spaces" (e.g. topological spaces, schemes, sets, etc.). We can see this in various ways. First, in any extensive category, an object over $X + Y$ is the same as an object over $X$ together with an object over $Y$. Second, if such a category has binary products, they distribute over binary coproducts:

$$ X \times (Y + Z) \cong X \times Y + X \times Z.$$

Third, in any extensive category, any object with a morphism to the initial object $0$ must be initial itself. All these are basic things we expect of a category of "spaces".

Furthermore, algebras of a Lawvere theory are a pretty good formalization of the general idea of "algebraic gadgets" (e.g. monoids, rings, $R$-algebras, etc.).

So, my question is an attempt at getting at the vaguer question "when does the opposite of a category of algebraic gadgets act like a category of spaces?"

One classic example is the Lawvere theory of commutative rings. Its category of algebras is CommRing, and the opposite of that is the category of affine schemes, which is extensive.

Another classic example is the Lawvere theory of commutative $R$-algebras for some commutative ring $R$. The opposite of its category of algebras is the category of affine schemes over $R$, which is again extensive.

But there should also be some more exotic examples, showing up e.g. in Durov's approach to algebraic geometry.

So, it would be nice to get more of a sense of the full range of examples: Lawvere theories whose category of algebras has extensive opposite.