internalization and categorical algebra
algebra object (associative, Lie, …)
The notion of action object or module object is the internalization of the notion of action/module of monoids (such as groups or rings) on sets (such as group representations or modules), into any monoidal category to yield a notion of actions of monoid objects (such as group objects or ring objects) on the objects of that category .
Given a monoidal category , and given a monoid in , then a left module object in over is
such that
(unitality) the following diagram commutes:
where is the left unitor isomorphism of .
(action property) the following diagram commutes
A module and the monoid it lies over do not necessarily belong to the same category, a fact suggested by the microcosm principle:
Given a monoidal category and an -module (also called -actegory) (supported by the monoidal action ), and given a monoid in , then a left module object in over is
such that
(unitality) the following diagram commutes:
where is the unitor of .
(action property) the following diagram commutes
where is the actor of .
(geometric actions)
A group object-action
internal to Sets is a G-set: a set equipped with a group action by a discrete group – this is the plain notion of a group action;
internal to TopologicalSpaces is a topological G-space: a topological group equipped with an action on a topological space by continuous functions;
internal to SmoothManifolds is a G-manifold: a Lie group equipped with an action on a smooth manifold by smooth functions;
internal to SimplicialSets is a simplicial group action,
(equivariant principal bundles)
A -equivariant principal bundle is an internal action of a group object internal to a category of internal -actions as in Example , such as G-sets/G-spaces/G-manifolds (an “equivariant group”) which satisfies, internally, principality and local triviality-condition.
(2-actions)
The notion of coherent action object in the 2-category Cat (of categories with functors and natural transformations) is a categorified notion of “action” (namely of monoidal categories), known as module categories (also: “actegories”), see also 2-module and n-module).
A module object in a symmetric monoidal category of spectra is a module spectrum over a ring spectrum.
Internal to just the stable homotopy category it is a homotopy module spectrum.
categorical algebra – contents
internalization and categorical algebra
algebra object (associative, Lie, …)
The general definition of internal actions seems to have first been formulated in:
following the general principle of internalization formulated in:
Review:
See also:
Last revised on December 9, 2022 at 05:46:13. See the history of this page for a list of all contributions to it.