I recently read about the marvelous bar construction. As far as I understand, it is something of a free resolution in an abstract setting. I'm wondering if it can be used to get topological insights on algebraic structures.
Recall the bar construction. A comonad $(T,\mu,\eta)$ on a category $\mathcal C$ is a comonoid in the strict monoidal category $\operatorname{End}(\mathcal C)$. Denoting $\Delta_a$ the augmented simplex category, it gives a strict strong monoidal functor $$ \Delta_a^{\rm op} \to \operatorname{End}(\mathcal C) $$ where $T$ is the image of the singleton $\mathbf 1$ of $\Delta_a$. It yields, for any $c \in \mathcal C$, an augmented simplicial object $$ \bar c \colon \Delta_a^{\rm op} \to \operatorname{End}(\mathcal C) \to \mathcal C $$ where the second arrow is the evaluation at $c$: this augmented simplicial object $\bar c$ is the bar construction over $c$.
Now suppose one has a forgetful functor $U \colon \mathcal C \to \mathsf{Set}$. It then gives an augmented simplicial set $U\bar c$. Does this simplicial set or its geometric realization give us any information about $c$? Is the homology/homotopy type of this object relevant to understand $c$ better?
The example I have in mind is when the comonad $T$ is the composite $F_LU_L$ of the free functor and forgetful functor associated to a Lawvere theory $L$ (and then $U$ is obviously $U_L$).