nLab locally graded category




The notion of locally graded category is a joint generalisation of enriched categories, actegories, and powered categories. The idea is that we grade the morphisms of a category by a monoidal category VV in such a way that identity morphisms have grade labeled by the unit object II and the composite of morphisms is graded by the tensor product of the separate grades.


Let (V,,I)(V, \otimes, I) be a monoidal category. For small VV, a locally VV-graded category is a category enriched over the presheaf category ([V op,Set],^,y V)([V^{op}, Set], \widehat\otimes, y_V), regarded with its monoidal structure given by Day convolution.

Locally indexed categories

The presheaf category [V op,Set][V^{op}, Set] is also cartesian monoidal. A category enriched in ([V op,Set],×,1)([V^{op}, Set], \times, 1) is called locally VV-indexed by Levy (2019).

When VV is cartesian monoidal, the two concepts coincide (Levy 2019).

As we have stated the definitions, this is trivial, but both concepts have more elementary reformulations that avoid size issues, for which this is a nontrivial theorem.

For example, with Levy (2019), slide 21 this gives an abstract proof of the standard fact that simplicially enriched categories can be viewed as those simplicial objects in Cat which take value in identity-on-objects functors (e.g. Riehl (2023), Prop. 1.2.3).


Locally graded categories were introduced as large VV-categories in:

See also

For the terminology we use see:

For the example of simplicially enriched category example see for instance:

Last revised on July 26, 2023 at 16:39:33. See the history of this page for a list of all contributions to it.