Locally Graded Category

Let $\mathcal{V}$ be a Monoidal Category. A locally graded category over $\mathcal{V}$ consists of:

• A set of objects $\mathcal{C}$
• For each, $x:\mathcal{V}$ and $a,b:\mathcal{C}$, a set $\mathcal{C}_{x}(a,b)$ of "$x$ graded morphisms".
• For all $f:\mathcal{C}_{y}(a,b)$ and $k:\mathcal{V}(x,y)$, a morphism $k^{*}(f):\mathcal{C}_{x}(a,b)$.
• Identity morphisms $id:\mathcal{C}_{I}(a,a)$
• For all $f:\mathcal{C}_{x}(a,b)$, $g:\mathcal{C}_{y}(b,c)$, a composite $f;g:\mathcal{C}_{x\otimes y}(a,c)$.

Moreover, we impose the following conditions:

• $id^{*}(f)=f$.
• $(k\circ l)^{*}(f)=l^{*}(k^{*}(f))$.
• $k^{*}(id)=id$
• $(k^{*}(f));(l^{*}(g))=(k\otimes l)^{*}(f;g)$
• $\lambda^{*}(id;f)=f$
• $\rho^{*}(f;id)=f$
• $\alpha^{*}(f;(g;h))=(f;g);h$

Locally graded categories generalize Enriched Categories and Actegories.

Reference

• https://ncatlab.org/nlab/show/locally%20graded%20category