maybe this question is trivial and, then this is the reason I've never seen this written. 


The motivation is to define internal $\infty$-groupoids (that are preferably) Kan fibrant and to see if Kan fibrancy is really a necessary ontological assumption. As I understand there are non-Kan fibrant models of smooth $\infty$-groupoids given by $\infty$-stacks over cartesian spaces (<http://ncatlab.org/nlab/show/smooth+infinity-groupoid>), however there are Kan fibrant versions for the smooth case too (<http://ncatlab.org/nlab/show/Kan-fibrant+simplicial+manifold>)


So the questions are more or less: 


1) Given a (closed) monoidal (model) category $\mathcal{V}$, is there an operad (that I will call $\text{Grpd}$) such that algebras over $\text{Grpd}$ are exactly the internal groupoids?


2)If the answer to 1) is yes. Let $\text{Grpd}\_{\infty}$ be the resolutions of $\text{Grpd}$. What's a $\text{Grpd}\_{\infty}$-algebra?


Thanks in advance


**EDIT**: Apparently, there's a misunderstanding regarding the notion of operad as noticed by Todd Trimble in the comments (I didn't know about this notation issues). I'm assuming the definition of operad given from page 23 to 30 of <http://www.cs.ox.ac.uk/people/bob.coecke/andrei.pdf>



Is there an operad that codifies groupoids?
The question (under the relaxed notion of operad in the original post) will be settled negatively if we show that the category $\mathrm{Grpd}$ (internal groupoids in $\mathrm{Set}$) isn't monadic over $\mathrm{Set}$ for *any* choice of underlying functor $U: \mathrm{Grpd} \to \mathrm{Set}$. (Note: I am using $\mathrm{Grpd}$ to denote the *category*, not the putative operad of the OP.) 


Categories that are monadic over $\mathrm{Set}$ enjoy strong exactness conditions: they are Barr-exact. In particular, a category that is monadic over $\mathrm{Set}$ in *some* way is a [regular category](http://ncatlab.org/nlab/show/regular+category), and I claim that $\mathrm{Grpd}$ is not regular. In fact I will give an example of a regular epi in $\mathrm{Grpd}$ that is not stable under pullback. 


Let's take a connected groupoid $G$ with two objects $0, 1$ whose automorphism group at either object is say $\mathbb{Z}$ (it won't matter much what the automorphism group is, but to be definite let's take this one). Let $A$ be the coproduct of two copies of $G$, and let $B$ be the quotient of $A$ obtained by identifying $1$ of the first copy with $0$ of the second. Then $B$ is a connected groupoid with three objects $0, 1, 2$ and with automorphism group the free group $\mathbb{Z} \ast \mathbb{Z}$, with the object $1$ being the identification of the two aforementioned objects of $A$. 


Let $C$ be the full subgroupoid of $B$ on the objects $0, 2$, with inclusion $i: C \to B$. Consider what you get by pulling back the quotient map $p$ along $i$: 


$$\begin{array}{ccc}
P &\to & A\\
q \downarrow & & \downarrow p \\
C & \stackrel{i}{\to} & B
\end{array}$$


The domain $P$ sits inside $A$ as the full subcategory whose objects are $0$ of the first copy of $G$ and $1$ of the second copy. There is no way $q: P \to C$ can be a quotient map, since the images of the two automorphism groups of $P$ under $q$ are disjoint (hence miss the arrows connecting $0$ to $2$ in $C$. 


**Edit:** Nor is $\mathrm{Grpd}$ described by a multisorted Lawvere theory. For example, the category of algebras of a $2$-sorted Lawvere theory is monadic over $\mathrm{Set}/2$, and such categories are again regular. One result along these lines is that if $\mathbf{D}$ is finitely complete and finitely cocomplete and coequalizers in $\mathbf{D}$ split, and $U: \mathbf{C} \to \mathbf{D}$ is monadic, then $\mathbf{C}$ is regular; see for example Proposition 4.6 [here](http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf#118). This applies in particular to any $\mathbf{D} = \mathrm{Set}/X$. 


