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Let $C$ be an $\infty$-category.

1. If $C$ has a zero object, then the unique functor $C \to pt$ is an ambidextrous adjoint (i.e. fits into an adjoint string periodic of order 2).

2. (If $C$ is semiadditive, then the diagonal functor $C \to C^2$ likewise is an ambidextrous adjunction.)

3. Let $C^{[1]}$ be the category of arrows of $C$. There is always an adjunction $cod \dashv id \dashv dom$ between the codomain functor, the functor which takes the identity arrow, and the domain functor. If $C$ has an initial object, this chain extends one step further to the left, and if $C$ has a terminal object, it extends one further to the right. If $C$ is pointed and has cokernels it extends an additional step to the left, and if $C$ has kernels it extends an additional step to the right.

If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 6 up to a twist by the suspension functor $\Sigma$.

4. Likewise, let $C^{[2]}$ be the category of composable pairs of arrows of $C$. The composition map $C^{[2]} \to C^{[1]}$ always fits into the middle of an adjoint string of length 5, which extends an additional step with intitial / terminal objects and again with kernels / cokernels.

If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 24 up to a twist by $\Sigma^3$.

Examples $(1,3,4)$ clearly suggest that if $C$ is stable, then between $C^{[n-1]}$ and $C^{[n]}$ we have a bi-infinite adjoint string, perhaps periodic (up to a shift) of order $(n+1)!$. But I'm not sure.

Various reindexing functors for functor categories into a stable $\infty$-category from various other finite categories also induce infinite adjoint strings. For instance,

5. the constant functor $C \to C^{\bullet \leftarrow \bullet \to \bullet}$ fits in a bi-infinite adjoint string which is periodic of order 4 up to a shift by $\Sigma$.

If $C$ is ambidextrous in the sense of (Hopkins and) Lurie, even more functor categories do this.

Another place this happens is in Grothendieck duality -- it turns out that an exact symmetric monoidal functor between nice tensor-triangulated categories either fits in an adjoint string of length 1, 3, 5, or bi-infinity. In the infinite case, it is periodic of order 6 up to a twist by the so-called "dualizing object".

Let $C$ be an $\infty$-category.

1. If $C$ has a zero object, then the unique functor $C \to pt$ is an ambidextrous adjoint (i.e. fits into an adjoint string periodic of order 2).

2. (If $C$ is semiadditive, then the diagonal functor $C \to C^2$ likewise is an ambidextrous adjunction.)

3. Let $C^{[1]}$ be the category of arrows of $C$. There is always an adjunction $cod \dashv id \dashv dom$ between the codomain functor, the functor which takes the identity arrow, and the domain functor. If $C$ has an initial object, this chain extends one step further to the left, and if $C$ has a terminal object, it extends one further to the right. If $C$ is pointed and has cokernels it extends an additional step to the left, and if $C$ has kernels it extends an additional step to the right.

   If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 6 up to a twist by the suspension functor $\Sigma$.

4. Likewise, let $C^{[2]}$ be the category of composable pairs of arrows of $C$. The composition map $C^{[2]} \to C^{[1]}$ always fits into the middle of an adjoint string of length 5, which extends an additional step with intitial / terminal objects and again with kernels / cokernels.

   If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 24 up to a twist by $\Sigma^3$.

Examples $(1,3,4)$ clearly suggest that if $C$ is stable, then between $C^{[n-1]}$ and $C^{[n]}$ we have a bi-infinite adjoint string, perhaps periodic (up to a shift) of order $(n+1)!$. But I'm not sure.

Various reindexing functors for functor categories into a stable $\infty$-category from various other finite categories also induce infinite adjoint strings. For instance,

5. the constant functor $C \to C^{\bullet \leftarrow \bullet \to \bullet}$ fits in a bi-infinite adjoint string which is periodic of order 4 up to a shift by $\Sigma$.

If $C$ is ambidextrous in the sense of (Hopkins and) Lurie, even more functor categories do this.

Another place this happens is in Grothendieck duality -- it turns out that an exact symmetric monoidal functor between nice tensor-triangulated categories either fits in an adjoint string of length 1, 3, 5, or bi-infinity. In the infinite case, it is periodic of order 6 up to a twist by the so-called "dualizing object".

Let $C$ be an $\infty$-category.

1. If $C$ has a zero object, then the unique functor $C \to pt$ is an ambidextrous adjoint (i.e. fits into an adjoint string periodic of order 2).

2. (If $C$ is semiadditive, then the diagonal functor $C \to C^2$ likewise is an ambidextrous adjunction.)

3. Let $C^{[1]}$ be the category of arrows of $C$. There is always an adjunction $cod \dashv id \dashv dom$ between the codomain functor, the functor which takes the identity arrow, and the domain functor. If $C$ has an initial object, this chain extends one step further to the left, and if $C$ has a terminal object, it extends one further to the right. If $C$ is pointed and has cokernels it extends an additional step to the left, and if $C$ has kernels it extends an additional step to the right.

If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 6 up to a twist by the suspension functor $\Sigma$.

4. Likewise, let $C^{[2]}$ be the category of composable pairs of arrows of $C$. The composition map $C^{[2]} \to C^{[1]}$ always fits into the middle of an adjoint string of length 5, which extends an additional step with intitial / terminal objects and again with kernels / cokernels.

If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 24 up to a twist by $\Sigma^3$.

Examples $(1,3,4)$ clearly suggest that if $C$ is stable, then between $C^{[n-1]}$ and $C^{[n]}$ we have a bi-infinite adjoint string, perhaps periodic (up to a shift) of order $(n+1)!$. But I'm not sure.

Various reindexing functors for functor categories into a stable $\infty$-category from various other finite categories also induce infinite adjoint strings. If $C$ is ambidextrous in the sense of (Hopkins and) Lurie, even more functor categories do this.

Another place this happens is in Grothendieck duality -- it turns out that an exact symmetric monoidal functor between nice tensor-triangulated categories either fits in an adjoint string of length 1, 3, 5, or bi-infinity. In the infinite case, it is periodic of order 6 up to a twist by the so-called "dualizing object".

Let $C$ be an $\infty$-category.

1. If $C$ has a zero object, then the unique functor $C \to pt$ is an ambidextrous adjoint (i.e. fits into an adjoint string periodic of order 2).

2. (If $C$ is semiadditive, then the diagonal functor $C \to C^2$ likewise is an ambidextrous adjunction.)

3. Let $C^{[1]}$ be the category of arrows of $C$. There is always an adjunction $cod \dashv id \dashv dom$ between the codomain functor, the functor which takes the identity arrow, and the domain functor. If $C$ has an initial object, this chain extends one step further to the left, and if $C$ has a terminal object, it extends one further to the right. If $C$ is pointed and has cokernels it extends an additional step to the left, and if $C$ has kernels it extends an additional step to the right.

If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 6 up to a twist by the suspension functor $\Sigma$.

4. Likewise, let $C^{[2]}$ be the category of composable pairs of arrows of $C$. The composition map $C^{[2]} \to C^{[1]}$ always fits into the middle of an adjoint string of length 5, which extends an additional step with intitial / terminal objects and again with kernels / cokernels.

If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 24 up to a twist by $\Sigma^3$.

Examples $(1,3,4)$ clearly suggest that if $C$ is stable, then between $C^{[n-1]}$ and $C^{[n]}$ we have a bi-infinite adjoint string, perhaps periodic (up to a shift) of order $(n+1)!$. But I'm not sure.

Various reindexing functors for functor categories into a stable $\infty$-category from various other finite categories also induce infinite adjoint strings. For instance,

5. the constant functor $C \to C^{\bullet \leftarrow \bullet \to \bullet}$ fits in a bi-infinite adjoint string which is periodic of order 4 up to a shift by $\Sigma$.

If $C$ is ambidextrous in the sense of (Hopkins and) Lurie, even more functor categories do this.

Another place this happens is in Grothendieck duality -- it turns out that an exact symmetric monoidal functor between nice tensor-triangulated categories either fits in an adjoint string of length 1, 3, 5, or bi-infinity. In the infinite case, it is periodic of order 6 up to a twist by the so-called "dualizing object".

Let $C$ be an $\infty$-category.

1. If $C$ has a zero object, then the unique functor $C \to pt$ is an ambidextrous adjoint (i.e. fits into an adjoint string of order 2).

2. (If $C$ is semiadditive, then the diagonal functor $C \to C^2$ likewise is an ambidextrous adjunction.)

3. Let $C^{[1]}$ be the category of arrows of $C$. There is always an adjunction $cod \dashv id \dashv dom$ between the codomain functor, the functor which takes the identity arrow, and the domain functor. If $C$ has an initial object, this chain extends one step further to the left, and if $C$ has a terminal object, it extends one further to the right. If $C$ is pointed and has cokernels it extends an additional step to the left, and if $C$ has kernels it extends an additional step to the right.

If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 6 up to a twist by the suspension functor $\Sigma$.

4. Likewise, let $C^{[2]}$ be the category of composable pairs of arrows of $C$. The composition map $C^{[2]} \to C^{[1]}$ always fits into the middle of an adjoint string of length 5, which extends an additional step with intitial / terminal objects and again with kernels / cokernels.

If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 24 up to a twist by $\Sigma^3$.

Examples $(1,3,4)$ clearly suggest that if $C$ is stable, then between $C^{[n-1]}$ and $C^{[n]}$ we have an infinite adjoint string, perhaps of order $(n+1)!$. But I'm not sure.

Various reindexing functors for functor categories into a stable $\infty$-category from various other finite categories also induce infinite adjoint strings. If $C$ is ambidextrous in the sense of (Hopkins and) Lurie, even more functor categories do this.

Let $C$ be an $\infty$-category.

1. If $C$ has a zero object, then the unique functor $C \to pt$ is an ambidextrous adjoint (i.e. fits into an adjoint string periodic of order 2).

2. (If $C$ is semiadditive, then the diagonal functor $C \to C^2$ likewise is an ambidextrous adjunction.)

3. Let $C^{[1]}$ be the category of arrows of $C$. There is always an adjunction $cod \dashv id \dashv dom$ between the codomain functor, the functor which takes the identity arrow, and the domain functor. If $C$ has an initial object, this chain extends one step further to the left, and if $C$ has a terminal object, it extends one further to the right. If $C$ is pointed and has cokernels it extends an additional step to the left, and if $C$ has kernels it extends an additional step to the right.

If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 6 up to a twist by the suspension functor $\Sigma$.

4. Likewise, let $C^{[2]}$ be the category of composable pairs of arrows of $C$. The composition map $C^{[2]} \to C^{[1]}$ always fits into the middle of an adjoint string of length 5, which extends an additional step with intitial / terminal objects and again with kernels / cokernels.

If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 24 up to a twist by $\Sigma^3$.

Examples $(1,3,4)$ clearly suggest that if $C$ is stable, then between $C^{[n-1]}$ and $C^{[n]}$ we have a bi-infinite adjoint string, perhaps periodic (up to a shift) of order $(n+1)!$. But I'm not sure.

Various reindexing functors for functor categories into a stable $\infty$-category from various other finite categories also induce infinite adjoint strings. If $C$ is ambidextrous in the sense of (Hopkins and) Lurie, even more functor categories do this.

Another place this happens is in Grothendieck duality -- it turns out that an exact symmetric monoidal functor between nice tensor-triangulated categories either fits in an adjoint string of length 1, 3, 5, or bi-infinity. In the infinite case, it is periodic of order 6 up to a twist by the so-called "dualizing object".