Weighted Limits in Categories Graded by Monoidal Categories

Categories graded by a monoidal category $\mathcal{V}$ generalize both $\mathcal{V}$-actegories and $\mathcal{V}$-enriched categories without requiring additional properties of $\mathcal{V}$. However, $\mathcal{V}$-graded categories are themselves enriched in the monoidal category $\hat{\mathcal{V}}$ of presheaves on $\mathcal{V}$. In this text, we define a notion of weighted limit for $\mathcal{V}$-graded categories, and show that $\mathcal{V}$-graded weighted limits are precisely the $\hat{\mathcal{V}}$-enriched weighted limits whose weights take on representable values. When $\mathcal{V}$ is biclosed and the $\mathcal{V}$-graded categories involved are $\mathcal{V}$-enriched, we recover precisely the familiar notion of $\mathcal{V}$-enriched weighted limit. We use $\mathcal{V}$-graded structure to define weighted limits in $\mathcal{V}$-actegories and in $\mathcal{V}$-categories for a non-biclosed monoidal $\mathcal{V}$. We develop both a convenient concrete formulation and an equivalent abstract description as $\mathcal{V}$-graded representations, and explore examples including $\mathcal{V}$-graded powers and conical limits.