The n-Category Café

While it’s not surprising, for the record I note that also Lazaroiu’s notion of morphisms of “$G$-graded categories with shifts” coincides the with notion of morphism of “categories with $G$-flow”:

Proposition What Lazaroiu calls a morphism of $G$-graded categories with shifts [Lazaroiu06, p. 7 below (2.5)] is exactly a morphism of categories with $G$-flow [Supercategories, def 2].

Proof. Lazroiu’s description (between (2.5) and (2.6))

a functor $F$ which intertwines $\gamma(g)$ and $\gamma'(g)$ and maps $s_a(g)$ into $s'_{F(a)}(g)$

is precisely the componentwise statement of the condition $$\array{ & \nearrow \searrow^{\mathrm{Id}} \\ C &\Downarrow^{s(g)}& C \\ & \searrow \nearrow^{\gamma(g)} \\ && \downarrow^F \\ && C' } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \array{ C \\ \downarrow^F \\ & \nearrow \searrow^{\mathrm{Id}} \\ C' &\Downarrow^{s'(g)}& C' \\ & \searrow \nearrow^{\gamma'(g)} }$$

Furthermore, the “skew category” $T[C]$ (bottom of p.7 of Lazaroiu’s article, apparently originally introduced in math.RA/0312214) which is obtained from a category $C$ with $G$-action $$\gamma : G \to \mathrm{Aut}(C)$$ is nothing but, I think, the weak coequalizer of this action in $\mathrm{Cat}$ $$C \stackrel{\stackrel{\mathrm{id}}{\to}}{\stackrel{\gamma(g)}{\to}} C \to T[C] \,.$$

This must be a well-known construction to category theorists, though I don’t know a canonical reference off the top of my head. I discussed this in a slightly different but analogous context in Universal Transition which later became section B.1 in my article with Konrad:

This coequalizer, which I claim is the same as $T[C]$, is generated from $C$ together with one new morphism $$a \stackrel{s_a(g)}{\to}\gamma(g)(a)$$ per object $a$of $C$ and element $g \in G$, subject to the relation $$\array{ a &\stackrel{f}{\to}& b \\ \downarrow^{s_a(g)} && \downarrow^{s_b(g)} \\ \gamma(g)(a) &\stackrel{\gamma(g)(f)}{\to}& \gamma(g)(b) }$$ for all $g$ and $f$ together with the obvious respect of the $s_\cdot(\cdot)$ for the group action.