Let $T$ be a monad on a category $\mathcal{C}$, and $A$ an algebra over $T$. The bar construction is a simplicial object in the category $\mathcal{C}^T$ of algebras which we can think of a sort of "resolution" of $A$. Some of the arrows look like the following diagram: $$ \begin{array}{ccc} \cdots TTA \rightrightarrows TA \to A \end{array} $$ (I unfortunately cannot draw more arrows here. See the link above for a better picture.)

Now, is such a simplicial object a Kan complex, or at least a quasicategory? Is there a filling condition for horns, in general? If not, what would be a counterexample?

Any reference would also be welcome.

Let $T$ be a monad on a concrete category $\mathcal{C}$, and $A$ an algebra over $T$. The bar construction is a simplicial object in the category $\mathcal{C}^T$ of algebras which we can think of a sort of "resolution" of $A$. Some of the arrows look like the following diagram: $$ \begin{array}{ccc} \cdots TTA \rightrightarrows TA \to A \end{array} $$ (I unfortunately cannot draw more arrows here. See the link above for a better picture.)

Now, is such a simplicial object a Kan complex, or at least a quasicategory? Is there a filling condition for horns, in general? If not, what would be a counterexample?

Any reference would also be welcome.