Given a monoid $C$ acting (from the right) on a set $M$, there is a bar construction giving a simplicial set or equivalently a translation/action category,

$$ N_\bullet (M\rtimes C)= \cdots M\times C^2 \Rrightarrow M\times C ⇉^s_t M, \quad \text{with } s(m,c)=mc, t(m,c)=m. $$ There is a natural morphism of simplicial sets $N_\bullet (M\rtimes C)\to N_\bullet (*\rtimes C)$. One may argue that the correct quotient of $M$ by the $C$ action is this bar construction (cf. Thomason’s homotopy colimit theorem).

Now consider weak monoidal category $C$ coherent action on a category $M$. That is, there is an action functor $M\times C\to M$, unit and associative up to coherent invertible natural transformations.

We imitate the bar construction $$ \cdots M\times C^2\Rrightarrow M\times C ⇉^s_t M. $$ But in this case simplicial identities hold up to invertible natural transformations. Jardine's supercoherence theorem implies that we obtain a pseudo-simplicial categories, that is a pseudo functor $\Delta^{op}\to Cats$.

Applying then the Grothendieck construction, we obtain a category which is a pseudo colimit of this simplicial diagram. This may be regard as the (homotopy) quotient of $M$ by the $C$ action.

There is another construction like the geometric nerve of a bicategory.

Viewing $C$ as a one object, say $*$, bicategory. Imaging the object of $M$ as arrows $0\leftarrow *$, arrows of $M$ as 2-cells. Then the action functor works like composition in a bicategory.

Now form a simplicial set with $X_0=M_0$, that is arrows $0\leftarrow *$. Let $X_1$ be the set of 2-commutative triangles with vertex $(0,*,*)$, that is the set of $(m, c, \alpha)$, where $m\in M, c\in C$ and $\alpha$ is an arrow in $M$ with target $mc$. Let $X_2$ be 2-commutaitve 3-simplex $(0,*,*,*)$, and so on. We obtain a simplicial set $X$.

(It is clear that if $M$ is a set and $C$ is a monoid, $X$ is the bar construction.)

My question: whether $X$ and the one given by Grothendieck construction have the same homotopy type? They both have a map to (the nverve of one obect bicategory) $C$, thus a commutative triangle up to homotopy?

Possible relevant result, it is shown that the geometric nerve and pseudo-nerve of a bicategory has the same homotopy type. However this does not answer the question directly.


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