nLab shift

Idea

Shifts are particular dynamical systems on spaces of sequences, or more generally on products, which intuitively act on the position of the components (also by possibly copying and discarding them), but do not on the internal values of the coordinates.

Shifts are ubiquitous in the theory of dynamical systems (and related fields, such as ergodic theory and information theory). From the point of view of category theory, they are a universal (right adjoint) way of forming dynamical systems from the objects of a category.

Definition

Let $X$ be a set, and consider the countably infinite product $X^\mathbb{N}$. The shift is the map which one can interpret as a “shift to the left”.

Equivalently, it is the product projection

$$X^\mathbb{N} \cong X \times X^\mathbb{N} \to X^\mathbb{N} .$$

Generalizations

In other categories

A similar definition of shift can be given in other categories with products, and more generally in any copy-discard category. In the field of dynamical systems, for example, the following categories are of interest:

• The category of compact metric spaces;
• The category of measurable spaces and measurable maps;
• The category of probability spaces and measure-preserving maps;
• The category of probability spaces and measure preserving Markov kernels.

(Outside the cartesian case, one needs a notion of infinite tensor product.)

For generic actions

One can generalize the definition as follows. Let $M$ be a monoid with an action on a finite set $S=\{1,\dots,n\}$, and denote the action by $m:S\to S$ for each $m\in M$.

Given an object $X$ in a category, consider the $S$-fold product $X^S$. We have an action of $M$ on $X^S$ given as follows. For each $m\in M$, we define the map $m^*:X^S\to X^S$ in terms of its product projections $\pi_i\circ m^*:X^S\to X$ given by

$$\pi_i\circ m^* \;\coloneqq\; \pi_{m(i)} .$$

Similarly, if $S$ is an infinite set, we define the action of $M$ on the infinite product $X^S$ whose finitary product projections are in the form above.

Even more generally, one can replace the product with an infinitary tensor product, and define the action analogously.

Examples

• The usual shift $\sigma:X^\mathbb{N}\to X^\mathbb{N}$ is induced by the action of $\mathbb{N}$ on itself via addition.
• The action of finite permutations on $X^\mathbb{N}$ is induced by the action of finite permutations on $\mathbb{N}$.

Universal property

Let $M$ be a monoid, and let $C$ be a category with products. Denote by $B M$ the delooping of $M$, and by $[B M, C]$ the functor category, i.e. the category of objects of $C$ equipped with an action of $M$, and equivariant maps as morphisms.

The forgetful functor $U:[B M,C]\to C$ has a right adjoint, given by the action of $M$ via shifts: on objects, it maps an object $X$ of $C$ to the product $X^M$, with the action of $M$ via shifts induced by the right-multiplication $M\times M\to M$.

On $C$, this adjunction induces a comonad, the stream comonad?.

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See also

• exchangeability, action
• stream comonad?

References

• Mike Behrisch, Sebastian Kerkhoff, Reinhard Pöschel, Friedrich Martin Schneider, Stefan Siegmund, Dynamical systems in categories. Applied Categorical Structures 25, 2017. (link)

• Paolo Perrone, Starting Category Theory, World Scientific, 2024. (website)

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category: probability