用户: Aripriner/Higher Algebra/monoids

定义 0.1. Let $\mathcal{C}$ be an $\infty$-category with finite products (in particular, $\mathcal{C}$ has a final object $\ast$). A cartesian monoid in $\mathcal{C}$ is a functor $X:{\Delta }^{op}\to \mathcal{C}$ such that:

1.	$X_0\simeq \ast$.
2.	It satisfies the Segal condition, i.e. the Segal maps $e_i:[1]\to [n],0\to i,1\to i+1$ defines an equivalence $$\begin{array}{r}{X}_n\simeq {\mathrm{Hom}}_{sAni}({\Delta }_n,X)\to {\mathrm{Hom}}_{sAni}(I_n,X)\simeq X_1{\times }_{X_0}{X}_1{\times }_{X_0}\cdots {\times }_{X_0}{X}_1=\prod _{i=1}^n{X}_1\end{array}$$Here $I_n$ are the collection of edges $i\to i+1$.

Let $\mathrm{Mon}(\mathcal{C})\subset s\mathcal{C}$ be the full sub-$\infty$-category spanned by cartesian monoids. If $\mathcal{C}=\mathrm{Ani}$, these are also called ${\mathbb{E}}_1$-monoids. For $\mathcal{C}={\mathrm{Cat}}_{\infty }$ we simply call them monoidal $\infty$-category (note that these can be encoded as cocartesian fibrations over ${\Delta }^{op}$).

命题 0.2. The completion functor $\mathrm{comp}:\mathrm{sAni}\to \mathrm{CSAni}$ restricts a fully faithful functor $$\begin{array}{r}\mathrm{comp}:\mathrm{Mon}(\mathrm{Ani})\to \ast /\mathrm{CSAni}\end{array}$$with essential image those pointed complete Segal spaces $(X,x)$ with ${\pi }_0(X_0)=\ast$.

In other words, monoids are really categories with one object (up to equivalence, but not up to contractible choices).
Under the equivalence ${\mathrm{Cat}}_{\infty }\simeq \mathrm{CSAni}$, we have the functor decomp corresponds to a functor $\ast /{\mathrm{Cat}}_{\infty }\to \mathrm{Mon}(\mathrm{Ani})$ which bacomes an equivalence when restricted to the full subcategory $(\ast /{\mathrm{Cat}}_{\infty }{)}_{{\ge }_1}\subseteq \ast /{\mathrm{Cat}}_{\infty }$ spanned by the object $\mathcal{C}$ with ${\pi }_0(core\mathcal{C})\simeq \ast$
Explicitly, this functor sends a pointed $\infty$-category $\{x\}\to \mathcal{C}$ to ${\mathrm{Hom}}_{\mathcal{C}}(x,x)$ in degree 1. We thus obtain

推论 0.3. If $\mathcal{C}$ is an $\infty$-category and $x\in \mathcal{C}$, then ${\mathrm{Hom}}_{\mathcal{C}}(x,x)$ carries a canonical structure of an ${\mathbb{E}}_1$-monoid.

定义 0.4. Let $\mathcal{C}$ be an $\infty$-category with finite porducts. A cartesian monoid $X$ in $\mathcal{C}$ is called a cartesian group (or ${\mathbb{E}}_1$-group in the case where $\mathcal{C}=\mathrm{Ani}$) if the map $$\begin{array}{r}(p{r}_1,\circ ):X_1\times X_1\to X_1\times X_1,(f,g)\to (f,f\circ g)\end{array}$$is an equivalence. We denote by $\mathrm{Grp}(\mathcal{C})\subseteq \mathrm{Mon}(\mathcal{C})$ the full subcategory of cartesian groups.

命题 0.5. The equivalence $\mathrm{Mon}(\mathrm{Ani})\simeq (\ast /{\mathrm{Cat}}_{\infty }{)}_{\ge 1}\simeq (\ast /\mathrm{CSAni}{)}_{{\pi }_0=\ast }$ restrict to equivalence $\mathrm{Grp}(\mathrm{Ani})\simeq (\ast /\mathrm{Ani}{)}_{\ge 1}\simeq (\ast /{\mathrm{sAni}}_{const}{)}_{{\pi }_0=\ast }$

In fact, the equivalnece $\mathrm{Grp}(\mathrm{Ani})\simeq (\ast /\mathrm{Ani}{)}_{\ge 1}$ is explicitly given as follows:$$\begin{array}{r}B=||={\mathrm{colim}}_{{\Delta }^{op}}:\mathrm{Grp}(\mathrm{Ani})\rightleftharpoons (\ast /\mathrm{Ani}{)}_{⩾1}:\Omega \end{array}$$To see the equivalence is given by the functors, recall that the composition$$\begin{array}{r}\mathrm{Mon}(\mathrm{Ani})\stackrel{\mathrm{comp}}{⟶}\mathrm{CSAni}\stackrel{\mathrm{comp}}{⟶}\mathrm{Ani}\end{array}$$sends $X\to |X^{\times }|$ by using $core(asscat(X))\simeq |X^{\times }|$ and unravelling the definition. If $X$ is an ${\mathbb{E}}_1$-group, the result of this composition is just the corresponding anima by the simplicial anima is constant. And we have $X^{\times }=X$. Thus $X\in \mathrm{Grp}(\mathrm{Ani})$ is sent to $|X|=BX$ as claimed.
Conversely, the map $\mathrm{decomp}:(\ast /\mathrm{Ani}{)}_{⩾1}\stackrel{\mathrm{decomp}}{⟶}\mathrm{Grp}(\mathrm{Ani})$ just sends $(K,k)$ to ${\mathrm{Hom}}_K(k,k)$ in degree 1. To show that the functor $\mathrm{decomp}$ acts like the loop functor, we just notice that ${\Omega }_k(K)\simeq {\mathrm{Hom}}_K(k,k)$.
In particular, we know that the functor $B$ and $\Omega$ are equivalences.

命题 0.6. The inclusion $\mathrm{Grp}(\mathrm{Ani})\subset \mathrm{Mon}(\mathrm{Ani})$ has a left adjoint called group completion and denoted $$\begin{array}{r}(-{)}^{\infty -\mathrm{grp}}:\mathrm{Mon}(\mathrm{Ani})\to \mathrm{Grp}(\mathrm{Ani})\end{array}$$It is given by $X^{\infty -\mathrm{grp}}\simeq \Omega BX$, where $B=||:\mathrm{Mon}(\mathrm{Ani})\to (\ast /\mathrm{Ani}{)}_{⩾1}$ just like the above.

证明. We know the the inclusion $\mathrm{Ani}\subseteq {\mathrm{Cat}}_{\infty }$ has left adjoint $||:{\mathrm{Cat}}_{\infty }\to \mathrm{Ani}$ and the adjunction can be extended to the slice category over the point $\ast$. Moreover the $(-{)}_{⩾1}$ parts on the both sides are preserved, thus $||$ descends to the left adjunction $||:(\ast /{\mathrm{Cat}}_{\infty }{)}_{⩾1}\to (\ast /\mathrm{Ani}{)}_{⩾1}$.
Now use the equivalence of Proposition 0.5, we get the left adjunction $||:\mathrm{Mon}(\mathrm{Ani})\to \mathrm{Grp}(\mathrm{Ani})$. To show it is the same as the functor above, we use the equivalence $|X|\simeq |asscat(X)|$ because both sides are in the $\infty$-category ${\mathrm{Fun}}^L(\mathcal{P}(\mathrm{Ani}),\mathrm{Ani})=\mathrm{Fun}({\Delta }^{op},\mathrm{Ani})$ and their restrictions on ${\Delta }^{op}$ are both sending ${\Delta }^n\to \ast$.

$\square$

推论 0.7. The functor $B$ and $\Omega$ form an adjunction $$\begin{array}{r}B:\mathrm{Mon}(\mathrm{Ani})\rightleftharpoons \ast /\mathrm{Ani}:\Omega \end{array}$$

Now we discuss smoe examples. We begin with free monoids and groups.

命题 0.8. The evaluation functor $e{v}_1:\mathrm{Mon}(\mathrm{Ani})\to \mathrm{Ani}$ has a left adjoint ${\mathrm{Free}}^{\mathrm{Mon}}:\mathrm{Ani}\to \mathrm{Mon}(\mathrm{Ani})$. And $$\begin{array}{r}{\mathrm{Free}}^{\mathrm{Mon}}(K{)}_1\simeq \coprod _{n⩾0}{K}^n\end{array}$$

命题 0.9. The evaluation functor ${\mathrm{ev}}_1:\mathrm{Grp}(\mathrm{Ani})\to \mathrm{Ani}$ has a left adjoint ${\mathrm{Free}}^{\mathrm{Grp}}:\mathrm{Ani}\to \mathrm{Grp}(\mathrm{Ani})$. It is given by the composite $$\begin{array}{rl}\mathrm{Ani}& \to \ast /\mathrm{Ani}\stackrel{\Sigma }{\to }\ast /\mathrm{Ani}\stackrel{\Omega }{\to }\mathrm{Grp}(\mathrm{Ani})\\ X& \mapsto X_{+}:=X\bigsqcup \ast \end{array}$$

注 0.10. Notice that the suspension functor $\Sigma$ is also a functor in the unpointed $\infty$-category $\mathrm{Ani}$. In fact, it does not matter whether this colimit is taken in $\mathrm{Ani}$ or $\ast /\mathrm{Ani}$ since $\ast /\mathrm{Ani}\to \mathrm{Ani}$ commutes with weakly contractible colimits (not any colimits by it does not preserve the initial object).
To check this, we need to show that the adjunction $$\begin{array}{r}{\mathrm{colim}}_{\mathcal{I}}:\mathrm{Fun}(\mathcal{I},\mathrm{Ani})\leftrightarrow \mathrm{Ani}:const\end{array}$$can be descended to the silce category. So we need to unit and counit are equivalence between the point $\ast$. When $\mathcal{I}$ is weakly contractible, ${\mathrm{colim}}_{\mathcal{I}}(const\ast )\simeq |\mathcal{I}|\simeq \ast$.
Similarly, once we have chose a basepoint $x\in X$, it does not matter to take the pullback ${\Omega }_{x}X$ in $\ast /\mathrm{Ani}$ or $\mathrm{Ani}$. This can be directly obtained by the fact that $\ast /\mathrm{Ani}\to \mathrm{Ani}$ has a left adjoint.