nLab locally graded category

Idea

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 $V$ in such a way that identity morphisms have grade labeled by the unit object $I$ and the composite of morphisms is graded by the tensor product of the separate grades.

Definition

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

Explicitly, a locally $V$-graded category $C$ comprises:

• a class $|C|$ of objects
• for all $v \in V$ and $c, c' \in |C|$, a set $C_v(c, c')$ of $v$-graded morphisms
• for every $v$-graded morphism $f : c \to c'$ and $p : v' \to v$, a $v'$-graded morphism $p^* f : c \to c'$
• a $I$-graded identity morphism $1_c$ for each object $c \in |C|$
• a $v \otimes v'$-graded composite morphism $g f : c \to c''$ for each $v$-graded morphism $f : c \to c'$ and $v'$-graded morphism $g : c' \to c''$

satisfying evident laws.

Locally indexed categories

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

When $V$ 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).

References

Locally graded categories were introduced as large $V$-categories in:

• Richard Wood, Indicial methods for relative categories, PhD thesis (1976) [hdl:10222/55465]

• Richard Wood, $V$-indexed categories, in Indexed Categories and Their Applications, Lecture Notes in Mathematics 661 (1978) 126-140 [doi:10.1007/BFb0061362]

See also

• Richard Wood, re: module for a category, message to categories mailing list (2003) [link]

For the terminology we use see:

• Paul Blain Levy, Locally graded categories, talk (2019) [slides:pdf, pdf]

• Dylan McDermott, Tarmo Uustalu, Flexibly graded monads and graded algebras, in: Mathematics of Program Construction MPC 2022, Lecture Notes in Computer Science 13544, Springer (2022) [doi:10.1007/978-3-031-16912-0_4]

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

• Emily Riehl, Prop. 1.2.3 in: Homotopy coherent structures, Expositions in Theory and Applications of Categories, 1 (2023) 1-31 [arXiv:1801.07404, tac:te1]

A generalisation of locally graded categories to skew-enrichment may be found in the following under the name skew-proactegories:

• Alexander Campbell, Skew-enriched categories, Applied Categorical Structures 26.3 (2018): 597-615.