Looping Formal Moduli via Hill-Lawson formula for pushouts

Recollection: The Hill-Lawson Formula

Working in $Sp$, the theorem of Hill-Lawson on $E_k$-pushouts (which makes sense in more general monoidal categories) says the following.

Given a sequence of two $E_k$ maps $A\stackrel{f}{\to }B\stackrel{g}{\to }R$ We can produce $E_k$-$B$-algebras by $E_k$-tensor:

• $B\to B\otimes A\otimes R$
• $B\to B\otimes R$ The theorem of Hill Lawson say the diagram

is a pushout of $E_k$-algebras under $B$. Here, $\mathbb{E}A$ is the free $E_{k+1}$ algebra on $A$.

Example Take our input data to be

$A\stackrel{\epsilon }{\to }\mathbb{S}\stackrel{Id}{\to }\mathbb{S}$ The theorem then says $\mathbb{S}{⨆}_A\mathbb{S}\simeq \mathbb{S}{⨂}_{\mathbb{E}A}\mathbb{S}$ That is, suspension in the category of augmented $E_k$ algebras, a priori a $E_k$ pushout, can be calculated as a $E_{k+1}$-tensor.

$E_k$-formal loop stacks

Consider the $E_k$-presheaf

$X_A:=Ma{p}_{Al{g}_{aug}^k}(\mathbb{D}(-),A)\in Fu{n}_{\ast }(Al{g}_{aug}^k,S)$ where $\mathbb{D}$ denotes $E_k$-Koszul duality. We calcuate, using the above formula

$\Omega X_A\simeq Ma{p}_{Al{g}_{aug}^k}(\Sigma \mathbb{D}(-),A)\simeq Ma{p}_{Al{g}_{aug}^k}(\mathbb{S}{\otimes }_{\mathbb{E}(\mathbb{D}(-))}\mathbb{S},A)$ This therefore gives an explicit algebraic description of the group stack struture via a coalgebra structure.

Example $\Omega X_A(\mathbb{S}\oplus M)\simeq Ma{p}_{Al{g}_{aug}^k}(\mathbb{S}{\otimes }_{Fre{e}^{k+1}({\Omega }^{n}M)}\mathbb{S},A)$