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:
Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Section 2.2 of: Fundamental algebraic geometry. Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. (2005) [MR2007f:14001, ISBN:978-0-8218-4245-4, lecture notes]
Saunders MacLane, Sections VII.4 in: Categories for the Working Mathematician, Springer (1979, 2nd ed.) [doi:10.1007/978-1-4757-4721-8]
John Michael Boardman, Algebraic objects in categories, Chapter 7 of: Stable Operations in Generalized Cohomology [pdf, pdf] in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology Oxford (1995) [doi:10.1016/B978-0-444-81779-2.X5000-7]
Magnus Forrester-Barker, Group Objects and Internal Categories [arXiv:math/0212065]
See also:
Francis Borceux, George Janelidze, Gregory Maxwell Kelly, p. 8 of: Internal object actions, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 (dml:249553)
Mark Hovey, Brooke Shipley, Jeff Smith, pp. 15 in: Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149-208 [arXiv:math/9801077, doi:10.1090/S0894-0347-99-00320-3]
Discussion that the category of module objects over a commutative monoid in a bicomplete closed symmetric monoidal category is itself bicomplete closed symmetric monoidal:
Mark Hovey, Brooke Shipley, Jeff Smith, Lemma 2.2.2 & 2.2.8 in: Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149-208 [arXiv:math/9801077, doi:10.1090/S0894-0347-99-00320-3]
Florian Marty, Prop. 1.2.14, 1.2.16, 1.2.17 in: Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative, Ph.D. Toulouse (2009) [theses:2009TOU30071, pdf]
Martin Brandenburg, Prop. 4.1.10: Tensor categorical foundations of algebraic geometry [arXiv:1410.1716]
See also:
Lecture notes:
Last revised on April 18, 2024 at 14:36:28. See the history of this page for a list of all contributions to it.