用户: Aripriner/Higher Algebra/infinty Spectra

定义 0.1. We say an $\infty$-category has finite limits if it has finite products and pullbacks. In this case, a supension functor $\Omega :\ast /\mathcal{C}\to \ast /\mathcal{C}$ can be defined and we put $$\begin{array}{r}\mathrm{Sp}(\mathcal{C}):=\underset{{\mathbb{N}}^{\mathrm{op}}}{\lim }(\cdots \stackrel{\Omega }{⟶}\ast /\mathcal{C}\stackrel{\Omega }{⟶}\ast /\mathcal{C})\end{array}$$called the $\infty$-category of spectrum objects in $\mathcal{C}$. If $\mathcal{C}=\mathrm{Ani}$, we obtain $\mathrm{Sp}=\mathrm{Sp}(\mathrm{Ani})$, the $\infty$-category of spectra.

命题 0.2. Suppose that $\mathcal{C}$ has finite limits. Then the forgettable functor $$\begin{array}{r}\mathrm{Sp}(\mathrm{CGrp}(\mathcal{C}))\to \mathrm{Sp}(\mathcal{C})\end{array}$$is an equivalence.

引理 0.3. For any $\infty$-category $\mathcal{D}$ with finite limits and a zero object $\ast \in \mathcal{D}$ the functor $\Omega :\mathcal{D}\to \mathcal{D}$ factors as $$\begin{array}{r}\Omega :\mathcal{D}\to \mathrm{Grp}(\mathcal{D})\stackrel{e{v}_1}{\to }\mathcal{D}\end{array}$$

This lemma is not obvious.

证明. Consider Yoneda embedding $\mathcal{D}\to \mathrm{Fun}({\mathcal{D}}^{\mathrm{op}},\ast /\mathrm{Ani})$(we can descend it to pointed case because the image of Yoneda embedding lies in the limit-preserving functors). On $\mathrm{Fun}({\mathcal{D}}^{\mathrm{op}},\ast /\mathrm{Ani})$ we have the loop functor $\Omega$ which is obtained by postcomposition with $\Omega :\ast /\mathrm{Ani}\to \ast /\mathrm{Ani}$. We have the diagram by we have known $\Omega$ has the lift for $\mathcal{D}=\ast /\mathrm{Ani}$

Thus we have $\Omega :\mathcal{D}\to \mathrm{Grp}(\mathrm{Fun}({\mathcal{D}}^{\mathrm{op}},\ast /\mathrm{Ani}))$. We can check that the image lies in $\mathrm{Grp}(\mathcal{D})$

$\square$

Because ${\mathbb{N}}^{\mathrm{op}}\subseteq {\mathbb{Z}}^{\mathrm{op}}$ if final we can also write $\mathrm{Sp}(\mathcal{C})={\lim }_{{\mathbb{Z}}^{\mathrm{op}}}(\cdots \stackrel{\Omega }{\to }\ast /\mathcal{C}\stackrel{\Omega }{\to }\ast /\mathcal{C}\cdots )$. This gives the evaluation map $$\begin{array}{r}{\Omega }^{\infty -i}:\mathrm{Sp}(\mathcal{C})\to \mathcal{C}\end{array}$$which maps to the i-th coordinate for all $i\in \mathbb{Z}$. In the case $\mathcal{C}=\mathrm{Ani}$, we can define homotopy groups for $X\in \mathrm{Sp}$:$$\begin{array}{r}{\pi }_i(X)={\pi }_0({\Omega }^{\infty +i}X)\end{array}$$If we think $X$ as a sequence $(X_i{)}_{i\in \mathbb{Z}}$, then we have ${\pi }_i(X)={\pi }_0(X_{-i})={\pi }_j(X_{j-i})$.

引理 0.4. A map $f:X\to Y$ of spectra is an equivalence in $\mathrm{Sp}$ iff it induces isomorphisms $f_{\ast }:{\pi }_{\ast }(X)\to {\pi }_{\ast }(Y)$ on homotopy groups.

命题 0.5.

1.	$\mathrm{Sp}(\mathcal{C})$ has finite limits.
2.	The only reason this does not work for colimits is that the degree colimits might not commute with the loop functor, so they do not define an element in $\mathrm{Sp}(\mathcal{C})$ If they commute, they it really define the colimit.
3.	$\Omega :\mathrm{Sp}(\mathcal{C})\to \mathrm{Sp}(\mathcal{C})$ is an equivalence.
4.	$\mathrm{Sp}(\mathcal{C})$ is additive.

Here I want to explain why $\mathrm{Sp}(\mathcal{C})$ has finite limits.

引理 0.6. $\mathrm{Hom}$ anima in the limit of some $\infty$-categories is the limit of $\mathrm{Hom}$ anima in each $\infty$-categories.

证明. Just notice that the diagram define ${\mathrm{Hom}}_{\mathcal{C}}(a,b)$ is a limit in ${\mathrm{Cat}}_{\infty }$ because the restriction map $\mathrm{Fun}({\Delta }^1,\mathcal{C})\to \mathrm{Fun}(\{0,1\},\mathcal{C})\simeq \mathcal{C}\times \mathcal{C}$ is an isofibration then the pullback in $sSet$ is a pullback in ${\mathrm{Cat}}_{\infty }$. Also the limit of a point $\ast$ is still a point because it is final in this category. So we just use the fact that limits commute.

$\square$

Combining this lemma and the fact that we can use $\mathrm{Hom}$ to detect limits, we know that $\mathrm{Sp}(\mathcal{C})$ has finite limits.
We also need a lemma to compute constant limits:

引理 0.7. Assume that $\mathcal{I}$ is a weakly contractible $\infty$-category, i.e. $|\mathcal{I}|\simeq \ast$. Consider the diagram $F:\mathcal{I}\to \mathcal{C}$. If the image of $F$ lies in $core(\mathcal{C})$, then $\lim F\simeq F(i)$ for any $i\in \mathcal{I}$. This also hold for colimits.

证明. We have $$\begin{array}{rl}{\mathrm{Hom}}_{\mathcal{C}}(c,\lim F)& \simeq {\mathrm{Hom}}_{\mathrm{Fun}(\mathcal{I},\mathcal{C})}(constc,F)\\ & \simeq {\mathrm{Hom}}_{\mathrm{Fun}(|\mathcal{I}|,\mathcal{C})}(constc,F)\\ & \simeq {\mathrm{Hom}}_{\mathrm{Fun}(\{i\},\mathcal{C})}(constc,F)\\ & \simeq {\mathrm{Hom}}_{\mathcal{C}}(c,F(i))\end{array}$$That is what we need.

$\square$

命题 0.8. The functor $$\begin{array}{r}{B}^{\infty }:\mathrm{CGrp}(\mathrm{Ani})\to \mathrm{Sp}\end{array}$$is fully faithful with essential image those $X\in \mathrm{Sp}$ with ${\pi }_i(X)=\ast$ for $i<0$. Also ${\pi }_i(B^{\infty }X)={\pi }_i(X_1)$ for all ${\mathbb{E}}_{\infty }$-group $X$.

例 0.9.

1.	If $\Omega :\ast /\mathcal{C}\to \ast /\mathcal{C}$ is an equivalence, then $\mathrm{Sp}(\mathcal{C})\simeq \mathcal{C}$. This is just lemma 0.7.
2.	$\mathrm{Sp}({\mathcal{D}}_{⩾0}(R))\simeq \mathcal{D}(R)$
3.	Consider the Elienberg-MacLane functor $K:D_{⩾0}(\mathbb{Z})\to \mathrm{Ani}$ which preserves limits. Hence we can lift it to $H:\mathcal{D}(\mathbb{Z})\simeq \mathrm{Sp}({\mathcal{D}}_{⩾0}(\mathbb{Z}))\to \mathrm{Sp}$. Thus we can define the Elienber-MacLane spectrum.

Also we have the reduced version ${\tilde{\mathcal{C}}}_{\bullet }:\ast /\mathrm{Ani}\leftrightarrow {\mathcal{D}}_{⩾0}(\mathbb{Z}):K$. We can lift it to$$\begin{array}{r}{C}_{\bullet }:\mathrm{Sp}\leftrightarrow \mathcal{D}(\mathbb{Z}):H\end{array}$$defined by the formula $C_{\bullet }(X)\simeq {\mathrm{colim}}_{i\in \mathbb{N}}{\tilde{\mathcal{C}}}_{\bullet }({\Omega }^{\infty -i}X)[-i]$. The transition maps are given by the natrual transformation ${\tilde{C}}_{\bullet }(\Omega -)\Rightarrow \tilde{C}(-)[-1]$(note that $\tilde{C}(\ast )\simeq 0$).
To show that $C_{\bullet }$ is really a functor, we just need to show it gives the adjoint functor:$$\begin{array}{rl}{\mathrm{Hom}}_{\mathcal{D}(\mathbb{Z})}(C_{\bullet }(X),D)& \simeq \lim {\mathrm{Hom}}_{{\mathcal{D}}_{⩾0}(\mathbb{Z})}({\tilde{C}}_{\bullet }({\Omega }^{\infty -i}X),({\tau }_{⩾-i}D)[i])\\ & \simeq \lim {\mathrm{Hom}}_{{\mathcal{D}}_{⩾0}(\mathbb{Z})}({\Omega }^{\infty -i}X,K(({\tau }_{⩾-i}D)[i])\\ & \simeq \lim {\mathrm{Hom}}_{{\mathcal{D}}_{⩾0}(\mathbb{Z})}({\Omega }^{\infty -i}X,{\Omega }^{\infty -i}HD)\\ & \simeq {\mathrm{Hom}}_{\mathrm{Sp}}(X,HD)\end{array}$$Now we define the homology and the cohomology of $X\in \mathrm{Sp}$ with coefficients in $A\in Ab$ to be $$\begin{array}{rl}{H}_{\ast }(X,A)& :=H_{\ast }(C_{\bullet }(X){\otimes }_{\mathbb{Z}}^{L}A)\\ H^{\ast }(X,A)& :=H_{\ast }(R{\mathrm{Hom}}_{\mathbb{Z}}(C_{\bullet }(X),A))\end{array}$$

命题 0.10. Suppose $\mathcal{C}$ is an $\infty$-category with a zero object $0$. TFAE:

1.	$\mathcal{C}$ has finite limits and $\Omega :\mathcal{C}\to \mathcal{C}$ is an equivalence.
2.	$\mathcal{C}$ has finite colimits and $\Sigma :\mathcal{C}\to \mathcal{C}$ is an equivalence.
3.	$\mathcal{C}$ has finite colimits and finite limits and a square in $\mathcal{C}$ is pushout iff it is a pullback.

Such $\infty$-category is called stable $\infty$-category.

An an immediate sequence, $\mathrm{Sp}(\mathcal{C})$ is stable for all $\infty$-category $\mathcal{C}$ with finite limits, in particular for $\mathrm{Sp}$ and $\mathcal{D}(R)$.

推论 0.11. For a functor $F:\mathcal{C}\to \mathcal{D}$ between stable $\infty$-categories, TFAE:

1.	$F$ preserves finite limits.
2.	$F$ preserves finite colimits.

定义 0.12.

1.	A functor is called left exact if it preserves finite limits.
2.	We denote by ${\mathrm{Cat}}_{\infty }^{lex}\subseteq {\mathrm{Cat}}_{\infty }$ the subcategory spanned by $\infty$-categories having finite limits and left exact functors. Similar for ${\mathrm{Cat}}_{\infty }^{rex}$.
3.	We denote ${\mathrm{Cat}}_{\infty }^{st}$ the subcategory spanned by all stable $\infty$-categories.

First, we show that $\mathrm{Sp}(-):{\mathrm{Cat}}_{\infty }^{lex}\to {\mathrm{Cat}}_{\infty }^{st}$ is a functor. This can be ensured by the lemma:

引理 0.13. Let ${\mathrm{Cat}}_{\infty }^{\mathcal{I}-ex}$ be the subcategory spanned by the $\infty$-categories having $\mathcal{I}$-limits and functors preserving them. Then the maps ${\lim }_{\mathcal{I}}:\mathrm{Fun}(\mathcal{I},\mathcal{C})\to \mathcal{C}$ assemble into a natural transformation $$\begin{array}{r}\underset{\mathcal{I}}{\lim }:\mathrm{Fun}(\mathcal{I},-)\Rightarrow (-)\end{array}$$of functors ${\mathrm{Cat}}_{\infty }^{\mathcal{I}-ex}\to {\mathrm{Cat}}_{\infty }$

定理 0.14. The functor $\mathrm{Sp}:{\mathrm{Cat}}_{\infty }^{lex}\to {\mathrm{Cat}}_{\infty }^{st}$ is right adjoint to the inclusion functor, the counit is given by ${\Omega }^{\infty }:\mathrm{Sp}(\mathcal{C})\to \mathcal{C}$. As a corollary, if $\mathcal{C}$ is stable, we have a canonical lift ${\mathrm{hom}}_{\mathcal{C}}:{\mathcal{C}}^{\mathrm{op}}\times \mathcal{C}\to \mathrm{Sp}$.

推论 0.15. The $\mathrm{Hom}$ functors on $\mathcal{D}(R)$ and $\mathcal{K}(R)$ have canonical refinements to $\mathrm{Sp}$, given by $$\begin{array}{r}{\mathrm{hom}}_{\mathcal{D}(R)}(C,D)\simeq H(R{\mathrm{Hom}}_R(C,D))\\ {\mathrm{hom}}_{\mathcal{K}(R)}(C,D)\simeq H({\underline{\mathrm{Hom}}}_R(C,D))\end{array}$$

Because $\mathrm{Sp}$ is stable, we know that it has finite colimits. To show that it is even cocomplete, we will write $\mathrm{Sp}$ as a Bousfield localisation of a suitable cocomplete $\infty$-category of prespectra.
Let $\mathcal{C}$ be an $\infty$-category with finite limits and a terminal object $\ast \in \mathcal{C}$. We let $$\begin{array}{r}\mathrm{PSp}(\mathcal{C})\subseteq \mathrm{Fun}({\mathbb{N}}^2,\ast /\mathcal{C})\end{array}$$be the full subcategory that "vanishing away from the diagonal". So the objects are like

To construct a fully faithful embedding $\mathrm{Sp}(\mathcal{C})\hookrightarrow \mathrm{PSp}(\mathcal{C})$, write ${\mathbb{N}}^2={\mathrm{colim}}_{n\in \mathbb{N}}{\mathbb{N}}_{⩽(n,n)}^2$ and consider the diagram

${\eta }_n$ comes from this: $$\begin{array}{r}{\eta }_n:\ast /\mathcal{C}\to \mathrm{Fun}({\mathbb{N}}_{⩽(n,n)}^2,\ast /\mathcal{C})\end{array}$$first send an object $X\in \ast /\mathcal{C}$ to a functor from ${\mathbb{N}}_{⩽(n,n)}^2-\{(0,0),\cdots ,(n-1,n-1)\}$ to $\ast /\mathcal{C}$ as follows:

Then we take the right Kan extension of it along ${\mathbb{N}}_{⩽(n,n)}^2-\{(0,0),\cdots ,(n-1,n-1)\}\hookrightarrow {\mathbb{N}}_{⩽(n,n)}^2$. These constructions are functorial. So we have ${\eta }_n$. Because the inclusion in the right Kan extension is full faithful, the right Kan extension functor is fully faithful. And the first functor is clearly fully faithful. So ${\eta }_n$ is fully faithful. Thus we have $\eta =\lim {\eta }_n$ is fully faithful. We can check that the image of $\eta$ lies in $\mathrm{Sp}(\mathcal{C})$ and we get the full faithful functor $\mathrm{Sp}(\mathcal{C})\hookrightarrow \mathrm{PSp}(\mathcal{C})$.
We can also obtain canonical equivalence $$\begin{array}{r}{\mathrm{Hom}}_{\mathrm{PSp}(\mathcal{C})}(X,Y)\simeq \underset{n\in {\mathbb{N}}^{\mathrm{op}}}{\lim }{\mathrm{Hom}}_{\ast /\mathcal{C}}(X_n,Y_n)\end{array}$$For all $X\in \mathrm{PSp}(\mathcal{C})$ and $Y\in \mathrm{Sp}(\mathcal{C})$.

命题 0.16. If $\ast /\mathcal{C}$ has sequential colimits and $\Omega :\ast /\mathcal{C}\to \ast /\mathcal{C}$ commutes with them, then the inclusion $\mathrm{Sp}(\mathcal{C})\hookrightarrow \mathrm{PSp}(\mathcal{C})$ has left adjoint $(-{)}^{sp}:\mathrm{PSp}(\mathcal{C})\to \mathrm{Sp}(\mathcal{C})$. On objects it is given by $$\begin{array}{r}{\Omega }^{\infty -i}(X^{sp})\simeq {\mathrm{colim}}_{n\in \mathbb{N}}{\Omega }^n{X}_{n+i}\end{array}$$

To use this proposition for the case $\mathcal{C}=\mathrm{Ani}$, we show

引理 0.17. The loop functor $\Omega$ for pointed anima commutes with sequential colimits. In particular, $\mathrm{Sp}$ is a Bousfield localisation of $\mathrm{PSp}$ and thus cocomplete.

Similarly to the construction of the embedding $\mathrm{Sp}(\mathcal{C})\hookrightarrow \mathrm{PSp}(\mathcal{C})$, we can have an adjunction $$\begin{array}{r}{\tilde{\Sigma }}^{\infty }:\ast /\mathcal{C}\leftrightarrow \mathrm{PSp}(\mathcal{C}):e{v}_{(0,0)}\end{array}$$by left Kan extension. Combining these adjunctions, we see that ${\Omega }^{\infty }:\mathrm{Sp}\to \ast /\mathrm{Ani}$ has a left adjoint ${\Sigma }^{\infty }=({\tilde{\Sigma }}^{\infty }{)}^{sp}:\ast /\mathrm{Ani}\to \mathrm{Sp}$. And we have $$\begin{array}{r}{\Omega }^{\infty }{\Sigma }^{\infty }(X,x)\simeq {\mathrm{colim}}_{n\in \mathbb{N}}{\Omega }^n{\Sigma }^n(X,x)\end{array}$$The left-hand side is sometimes denoted $Q(X,x)$.
We denote the composite $$\begin{array}{r}\mathbb{S}[-]:\mathrm{Ani}\stackrel{(-{)}_{+}}{⟶}\ast /\mathrm{Ani}\stackrel{{\Sigma }^{\infty }}{⟶}\mathrm{Sp}\end{array}$$In particular, we have $\mathbb{S}:=\mathbb{S}[\ast ]$, the sphere spectrum. We can show that $\mathbb{S}$ represents the functor ${\Omega }^{\infty }:\mathrm{Sp}\to \mathrm{Ani}$.