geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
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.
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
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:
(Outside the cartesian case, one needs a notion of infinite tensor product.)
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
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.
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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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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Last revised on July 21, 2024 at 15:42:37. See the history of this page for a list of all contributions to it.