[05XD]
Lemma 2.2.12 . Let π \mathcal{C} β \infty β \infty X β π X\in\mathcal{C} m β β m\in\mathbb{N}
End π red β‘ ( X ) β ( m ) β Map π β‘ ( X β m , X ) β Ο β Map π β‘ ( X β m , X ) , \operatorname{End}_{\mathcal{C}}^{\operatorname{\scriptsize{red}}}\left(X\right)\left(m\right)\to\operatorname{Map}_{\mathcal{C}}\left(X^{\otimes m},X\right)\xrightarrow{\sigma^{*}}\operatorname{Map}_{\mathcal{C}}\left(X^{\sqcup m},X\right),
where the fiber is taken over the fold map β : X β m β X \nabla\colon X^{\sqcup m}\to X
[05XE]
Proof. By 2.2.9 β \infty
End π red β‘ ( X ) \textstyle{\operatorname{End}_{\mathcal{C}}^{\operatorname{\scriptsize{red}}}\left(X\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} π \textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces} πΌ β \textstyle{\mathbb{E}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces} π Β― β , \textstyle{\underline{\mathcal{C}}_{\sqcup},}
which, by 2.2.11
End π red β‘ ( X ) β ( m ) \textstyle{\operatorname{End}_{\mathcal{C}}^{\operatorname{\scriptsize{red}}}\left(X\right)\left(m\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Mul π β‘ ( X ( m ) , X ) \textstyle{\operatorname{Mul}_{\mathcal{C}}\left(X^{\left(m\right)},X\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces} πΌ β β ( m ) \textstyle{\mathbb{E}_{\infty}\left(m\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Mul π Β― β β‘ ( X ( m ) , X ) . \textstyle{\operatorname{Mul}_{\underline{\mathcal{C}}_{\sqcup}}\left(X^{\left(m\right)},X\right).}
The bottom map is the map Ξ 0 β Map π β‘ ( X β m , X ) \Delta^{0}\to\operatorname{Map}_{\mathcal{C}}\left(X^{\sqcup m},X\right) πΌ β = ( Ξ 0 ) β β π Β― β \mathbb{E}_{\infty}=\left(\Delta^{0}\right)_{\sqcup}\to\underline{\mathcal{C}}_{\sqcup} Ο : X β m β X β m \sigma\colon X^{\sqcup m}\to X^{\otimes m}
( β ) β : πππ β β ππ© β un : ( β ) Β― . \left(-\right)_{\sqcup}\colon\mathbf{Cat}_{\infty}\leftrightarrows\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}\colon\underline{\left(-\right)}.
β