symmetric monoidal (∞,1)-category of spectra
By module category may be meant
a category equipped with an action of a monoidal category (more commonly called actegory),
a category of modules of a monoid (e.g. an associative algebra),
a linear category equipped with an action of a tensor linear category (basically the first bullet point but in the -enriched setting).
Here we consider the first sense. For the second, see at category of modules.
The collection of module categories over a monoidal category forms a 2-category of module categories.
Let be a monoidal category and its delooping as a bicategory. A (left) module category is then simply a 2-functor .
Written out, this amounts to:
Further expanding this definition, we have the following data:
An obvious example is given by a monoidal category, which has an action on itself by left multiplication.
For instance
G. Janelidze, G.M. Kelly, A note on actions of a monoidal category, Theory and Applications of Categories, Vol. 9, No. 4, 2001, pp. 61–91, pdf.
Robert Gordon, A.J. Power, Enrichment through variation, J. Pure Appl. Algebra 120 (1997), 167-185.
Chris Douglas, Chris Schommer-Pries, Noah Snyder, Dualizable tensor categories (arXiv:1312.7188)
Last revised on September 27, 2023 at 13:25:53. See the history of this page for a list of all contributions to it.