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There is a striking similarity between [[displayed categories]], enriched categories,

and actegories. All three of these concepts take the basic algebra of a category and

replace the hom *sets* with some other notion of morphism graded by some base category

$B$, and then equip those morphisms with an action of $B$.

For a displayed category $\cE \liesover \cB$, our $\cB$-graded morphisms

are the displayed hom sets $\cE_{u}(X', Y')$ over indexed over a $u : \cB(X,Y)$.

The action of $\cB$ is a bit hard to see initially, but becomes painfully

clear once you've gotten your hands dirty: it is given by transport of hom families.

For an enriched category $\cC$, our $\cV$-graded morphisms are generalized

elements $\cV(\Gamma, \cC(X,Y))$ for $\Gamma : \cV$. Moreover, $\cV$ acts on

graded morphisms $\cV(\Gamma, \cC(X,Y))$ via precomposition with a morphism

$\cV(\Delta, \Gamma)$.

Finally, for an actegory $\cC$ equipped with an action $\oslash : \cV \times \cC \to \cC$,

the appropriate notion of $V$-graded morphism is a sort of sort of "$\cV$-generalized morphism"

$\cC(\Gamma \oslash X, Y)$ where $\Gamma : \cV$. The action of $\cV$ on these morphisms is given

by functoriality of $\oslash$ and precomposition.

**Locally graded categories** simultaneously generalize these three related notions

by combining the indexing of a displayed category with a sort of "directed transport"

operation that lets us move between indices. Explicitly, a locally graded category $\cE$

over a [[prebicategory]] $\cB$ consists of:

- A family of objects $\cC_0 : \cB_0 \to \ty$ indexed by the objects of $\cB$.

- A family of morphisms $\cC_1 : \cB_1(X,Y) \to \cC_0(X) \to \cC_0(Y) \to \set$ indexed

by the 1-cells of $\cB$.

- Identity and composite morphisms indexed over the identity 1-cell and composites of 1-cells.

- An action $[-] : \cB_2(u,v) \to \cC_1(v,X',Y') \to \cC_1(u,X',Y')$ of 2-cells of $\cB$

on morphisms of $\cC$.

We also require that composition is associative and unital once

suitably adjusted by an associator or unitor.

Finally, we require that the action of 2-cells is functorial, and that

identities and composites are sufficiently natural.