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I've seen many great positive answers to this question, providing examples of exotic internal homs. In this answer, I would like to show obstructions to the exoticism of the monoidal closed structure. I think that these examples will make this question and all the provided answers even more interesting. Nothing shapes like a boundary.

The most classical result in this direction is the monoidal structure on the category $\mathsf{Top}$ of topological spaces.

Prop. 7.1.1 Handbook of categorical Algebra 2, Borceux shows that the internal hom of any simmetric monoidal structure in Top has to match with the cartesian structure at least on the level of the underlying set.

This result highly depends on the fact that whatever monoidal closed structure $(\mathsf{I}, \otimes, [\_,\_])$ you have on a category $\mathcal{A}$, the $\mathsf{I}$-points of the internal hom recover the external hom,

$$\mathcal{A}(\mathsf{I}, [A,B]) \cong \mathcal{A}(A,B).$$

This observation spots an entanglement between the internal and the external logic of the category that unveil some rigidities of the monoidal structure.

Along those lines some research has been developed in the direction of showing that there exists some obstruction in admitting a monoidal biclosed structure.

Topological categories with many symmetric monoidal closed structures, by Kelly and Rossi, show that there exist topological categories which admit a proper class of symmetric monoidal closed structures.

Algebraic categories with few monoidal biclosed structures or none by Kelly, Foltz and Lair, goes in the other direction, proving (among other stuff) the two following theorems.

Prop. If an equational variety admits a monoidal biclosed structure, every idempotent algebra is self-commuting.

Prop. The categories of magmas, of semigroups, of magmas with identity, of monoids, of groups, of rings, and of commutative rings, admit no monoidal biclosed structures whatsoever; the category of abelian groups admits none but the classical one, and similarly for abelian monoids; and that the category of small categories admits exactly two, each symmetric, one being the classical Cartesian closed structure.

The last paper is very much inspired by the Czech school, among the very influential papers let me mention

• A. Pultr, Extending tensor products to structures of closed categories, Comm. Math. Univ. Carolinae 13 (1972) 599-616.

• A. Pultr, Closed categories of models of Gabriel theories (manuscript, Charles Univ. Prague, 1973).
• J. Rosicky, One obstruction for closedness, Comm. Math. Univ. Carolinae 18 (1977). 311-318.

If you have other examples that prove how having a closed monoidal structure imposes some rigidity on the underlying category, please contribute to this question with a comment.

I've seen many great positive answers to this question, providing examples of exotic internal homs. In this answer, I would like to show obstructions to the exoticism of the monoidal closed structure. I think that these examples will make this question and all the provided answers even more interesting. Nothing shapes like a boundary.

The most classical result in this direction is the monoidal structure on the category $\mathsf{Top}$ of topological spaces.

Prop. 7.1.1 Handbook of categorical Algebra 2, Borceux shows that the internal hom of any simmetric monoidal structure in Top has to match with the cartesian structure at least on the level of the underlying set.

This result highly depends on the fact that whatever monoidal closed structure $(\mathsf{I}, \otimes, [\_,\_])$ you have on a category $\mathcal{A}$, the $\mathsf{I}$-points of the internal hom recover the external hom,

$$\mathcal{A}(\mathsf{I}, [A,B]) \cong \mathcal{A}(A,B).$$

This observation spots an entanglement between the internal and the external logic of the category that unveil some rigidities of the monoidal structure.

Along those lines some research has been developed in the direction of showing that there exists some obstruction in admitting a monoidal biclosed structure.

Topological categories with many symmetric monoidal closed structures, by Kelly and Rossi, show that there exist topological categories which admit a proper class of symmetric monoidal closed structures.

Algebraic categories with few monoidal biclosed structures or none by Kelly, Foltz and Lair, goes in the other direction, proving (among other stuff) the two following theorems.

Prop. If an equational variety admits a monoidal biclosed structure, every idempotent algebra is self-commuting.

Prop. The categories of magmas, of semigroups, of magmas with identity, of monoids, of groups, of rings, and of commutative rings, admit no monoidal biclosed structures whatsoever; the category of abelian groups admits none but the classical one, and similarly for abelian monoids; and that the category of small categories admits exactly two, each symmetric, one being the classical Cartesian closed structure.

The last paper is very much inspired by the Czech school, among the very influential papers let me mention

• A. Pultr, Extending tensor products to structures of closed categories, Comm. Math. Univ. Carolinae 13 (1972) 599-616.

• A. Pultr, Closed categories of models of Gabriel theories (manuscript, Charles Univ. Prague, 1973).
• J. Rosický, One obstruction for closedness, Comm. Math. Univ. Carolinae 18 (1977). 311-318.

If you have other examples that prove how having a closed monoidal structure imposes some rigidity on the underlying category, please contribute to this question with a comment.

I've seen many great positive answers to this question, providing examples of exotic internal homs. In this answer, I would like to show obstructions to the exoticism of the monoidal closed structure. I think that these examples will make this question and all the provided answers even more interesting. Nothing shapes like a boundary.

The most classical result in this direction is the monoidal structure on the category $\mathsf{Top}$ of topological spaces.

Prop. 7.1.1 Handbook of categorical Algebra 2, Borceux shows that the internal hom of any simmetric monoidal structure in Top has to match with the cartesian structure at least on the level of the underlying set.

This results highly depends on the fact that whatever monoidal closed structure $(\mathsf{I}, \otimes, [\_,\_])$ you have on a category $\mathcal{A}$, the $\mathsf{I}$-points of the internal hom recover the external hom,

$$\mathcal{A}(\mathsf{I}, [A,B]) \cong \mathcal{A}(A,B).$$

This observation spots an entanglement between the internal and the external logic of the category that unveil some rigidities of the monoidal structure.

Along those lines some research has been developed in the direction of showing that there exists some obstruction in admitting a monoidal biclosed structure.

Topological categories with many symmetric monoidal closed structures, by Kelly and Rossi, show that there exist topological categories which admit a proper class of symmetric monoidal closed structures.

Algebraic categories with few monoidal biclosed structures or none by Kelly, Foltz and Lair, goes in the other direction, proving (among other stuff) the two following theorems.

Prop. If an equational variety admits a monoidal biclosed structure, every idempotent algebra is self-commuting.

Prop. The categories of magmas, of semigroups, of magmas with identity, of monoids, of groups, of rings, and of commutative rings, admit no monoidal biclosed structures whatsoever; the category of abelian groups admits none but the classical one, and similarly for abelian monoids; and that the category of small categories admits exactly two, each symmetric, one being the classical Cartesian closed structure.

The last paper is very much inspired by the Czech school, among the very influential papers let me mention

• A. Pultr, Extending tensor products to structures of closed categories, Comm. Math. Univ. Carolinae 13 (1972) 599-616.

• A. Pultr, Closed categories of models of Gabriel theories (manuscript, Charles Univ. Prague, 1973).
• J. Rosicky, One obstruction for closedness, Comm. Math. Univ. Carolinae 18 (1977). 311-318.

If you have other examples that prove how having a closed monoidal structure imposes some rigidity on the underlying category, please contribute to this question with a comment.

I've seen many great positive answers to this question, providing examples of exotic internal homs. In this answer, I would like to show obstructions to the exoticism of the monoidal closed structure. I think that these examples will make this question and all the provided answers even more interesting. Nothing shapes like a boundary.

The most classical result in this direction is the monoidal structure on the category $\mathsf{Top}$ of topological spaces.

Prop. 7.1.1 Handbook of categorical Algebra 2, Borceux shows that the internal hom of any simmetric monoidal structure in Top has to match with the cartesian structure at least on the level of the underlying set.

This result highly depends on the fact that whatever monoidal closed structure $(\mathsf{I}, \otimes, [\_,\_])$ you have on a category $\mathcal{A}$, the $\mathsf{I}$-points of the internal hom recover the external hom,

$$\mathcal{A}(\mathsf{I}, [A,B]) \cong \mathcal{A}(A,B).$$

This observation spots an entanglement between the internal and the external logic of the category that unveil some rigidities of the monoidal structure.

Along those lines some research has been developed in the direction of showing that there exists some obstruction in admitting a monoidal biclosed structure.

Topological categories with many symmetric monoidal closed structures, by Kelly and Rossi, show that there exist topological categories which admit a proper class of symmetric monoidal closed structures.

Algebraic categories with few monoidal biclosed structures or none by Kelly, Foltz and Lair, goes in the other direction, proving (among other stuff) the two following theorems.

Prop. If an equational variety admits a monoidal biclosed structure, every idempotent algebra is self-commuting.

Prop. The categories of magmas, of semigroups, of magmas with identity, of monoids, of groups, of rings, and of commutative rings, admit no monoidal biclosed structures whatsoever; the category of abelian groups admits none but the classical one, and similarly for abelian monoids; and that the category of small categories admits exactly two, each symmetric, one being the classical Cartesian closed structure.

The last paper is very much inspired by the Czech school, among the very influential papers let me mention

• A. Pultr, Extending tensor products to structures of closed categories, Comm. Math. Univ. Carolinae 13 (1972) 599-616.

• A. Pultr, Closed categories of models of Gabriel theories (manuscript, Charles Univ. Prague, 1973).
• J. Rosicky, One obstruction for closedness, Comm. Math. Univ. Carolinae 18 (1977). 311-318.

If you have other examples that prove how having a closed monoidal structure imposes some rigidity on the underlying category, please contribute to this question with a comment.

I've seen many great positive answers to this question, providing examples of exotic internal homs. In this answer, I would like to show obstructions to the exoticism of the monoidal closed structure. I think that these examples will make this question and all the provided answers even more interesting. Nothing shapes like a boundary.

The most classical result in this direction is the monoidal structure on the category $\mathsf{Top}$ of topological spaces.

Prop. 7.1.1 Handbook of categorical Algebra 2, Borceux shows that the internal hom of any simmetric monoidal structure in Top has to match with the cartesian structure at least on the level of the underlying set.

This results highly depends on the fact that whatever monoidal closed structure $(\mathsf{I}, \otimes, [\_,\_])$ you have on a category $\mathcal{A}$, the $\mathsf{I}$-points of the internal hom recover the external hom,

$$\mathcal{A}(\mathsf{I}, [A,B]) \cong \mathcal{A}(A,B).$$

This observation spots an entanglement between the internal and the external logic of the category that unveil some rigidities of the monoidal structure.

Along those lines some research has been developed in the direction of showing that there exists some obstruction in admitting a monoidal biclosed structure.

Topological categories with many symmetric monoidal closed structures, by Kelly and Rossi, show that there exist topological categories (of quasi-topological spaces) which admit a proper class of symmetric monoidal closed structures.

Algebraic categories with few monoidal biclosed structures or none by Kelly, Foltz and Lair, goes in the other direction, proving (among other stuff the two following theorems.

Prop. If an equational variety admits a monoidal biclosed structure, every idempotent algebra is self-commuting.

Prop. The categories of magmas, of semigroups, of magmas with identity, of monoids, of groups, of rings, and of commutative rings, admit no monoidal biclosed structures whatsoever; the category of abelian groups admits none but the classical one, and similarly for abelian monoids; and that the category of small categories admits exactly two, each symmetric, one being the classical Cartesian closed structure.

The last paper is very much inspired by the Czech school, among the very influential papers let me mention

• A. Pultr, Extending tensor products to structures of closed categories, Comm. Math. Univ. Carolinae 13 (1972) 599-616.

• A. Pultr, Closed categories of models of Gabriel theories (manuscript, Charles Univ. Prague, 1973).
• J. Rosicky, One obstruction for closedness, Comm. Math. Univ. Carolinae 18 (1977). 311-318.

If you have other examples that prove how having a closed monoidal structure imposes some rigidity on the underlying category, please contribute to this question with a comment.

I've seen many great positive answers to this question, providing examples of exotic internal homs. In this answer, I would like to show obstructions to the exoticism of the monoidal closed structure. I think that these examples will make this question and all the provided answers even more interesting. Nothing shapes like a boundary.

The most classical result in this direction is the monoidal structure on the category $\mathsf{Top}$ of topological spaces.

Prop. 7.1.1 Handbook of categorical Algebra 2, Borceux shows that the internal hom of any simmetric monoidal structure in Top has to match with the cartesian structure at least on the level of the underlying set.

This results highly depends on the fact that whatever monoidal closed structure $(\mathsf{I}, \otimes, [\_,\_])$ you have on a category $\mathcal{A}$, the $\mathsf{I}$-points of the internal hom recover the external hom,

$$\mathcal{A}(\mathsf{I}, [A,B]) \cong \mathcal{A}(A,B).$$

This observation spots an entanglement between the internal and the external logic of the category that unveil some rigidities of the monoidal structure.

Along those lines some research has been developed in the direction of showing that there exists some obstruction in admitting a monoidal biclosed structure.

Topological categories with many symmetric monoidal closed structures, by Kelly and Rossi, show that there exist topological categories which admit a proper class of symmetric monoidal closed structures.

Algebraic categories with few monoidal biclosed structures or none by Kelly, Foltz and Lair, goes in the other direction, proving (among other stuff) the two following theorems.

Prop. If an equational variety admits a monoidal biclosed structure, every idempotent algebra is self-commuting.

Prop. The categories of magmas, of semigroups, of magmas with identity, of monoids, of groups, of rings, and of commutative rings, admit no monoidal biclosed structures whatsoever; the category of abelian groups admits none but the classical one, and similarly for abelian monoids; and that the category of small categories admits exactly two, each symmetric, one being the classical Cartesian closed structure.

The last paper is very much inspired by the Czech school, among the very influential papers let me mention

• A. Pultr, Extending tensor products to structures of closed categories, Comm. Math. Univ. Carolinae 13 (1972) 599-616.

• A. Pultr, Closed categories of models of Gabriel theories (manuscript, Charles Univ. Prague, 1973).
• J. Rosicky, One obstruction for closedness, Comm. Math. Univ. Carolinae 18 (1977). 311-318.

If you have other examples that prove how having a closed monoidal structure imposes some rigidity on the underlying category, please contribute to this question with a comment.