Subsection 4.2 (0M3G) · ScalingStacks

4.2 Truncatedness and Connectedness[0M3G]

We recall the following definition from classical homotopy theory:

Definition 4.2.1. For $d\geq-2$ , a map $f\colon X\to Y$ of spaces is called $d$ -truncated if all of its homotopy fibers are $d$ -truncated spaces (3.1.1).

Using this definition, one can define a general notion of $d$ -truncatedness in an $\infty$ -category.

[0M31]

Definition 4.2.2. (T.5.5.6.1) For $d\geq-2$ , a map $f\colon X\to Y$ in an $\infty$ -category $\mathcal{C}$ is called $d$ -truncated, if for every $Z\in\mathcal{C}$ the induced map

\operatorname{Map}\left(Z,X\right)\to\operatorname{Map}\left(Z,Y\right)

is a $d$ -truncated map of spaces. An object $X$ is $d$ -truncated, if the map $X\to\text{pt}_{\mathcal{C}}$ is $d$ -truncated. We denote by $\tau_{\leq d}\mathcal{C}$ the full subcategory of $\mathcal{C}$ spanned by the $d$ -truncated objects. When $\mathcal{C}$ is presentable, by T.5.5.6.21 the $\infty$ -category $\tau_{\leq d}\mathcal{C}$ is itself presentable and by T.5.5.6.18, the inclusion $\tau_{\leq d}\mathcal{C}\hookrightarrow\mathcal{C}$ has a left adjoint $\tau_{\leq d}^{\mathcal{C}}\colon\mathcal{C}\to\tau_{\leq d}\mathcal{C}$ .

[05YH]

Remark 4.2.3. It is not difficult to show that $\tau_{\leq d}$ extends to a functor from the $\infty$ -category of presentable $\infty$ -categories to the full subcategory spanned by presentable essentially $\left(d+1\right)$ -categories and that it is left adjoint to the inclusion. The maps $\tau_{\leq d}^{\mathcal{C}}$ can be taken to be the components of the unit transformation (this essentially follows from T.5.5.6.22), but we shall not need this.

We now turn to discuss the dual notion of $n$ -connectedness.

[0M32]

Definition 4.2.4. For $n\geq-2$ , a map $f\colon A\to B$ in an $\infty$ -category $\mathcal{C}$ is $n$ -connected if it is left orthogonal to every $n$ -truncated map; ie for every commutative square $q\colon\Delta^{1}\times\Delta^{1}\to\mathcal{C}$ ,

in which $g$ is $n$ -truncated, $L\left(q\right)$ is contractible. An object $A\in\mathcal{C}$ is called $n$ -connected if $A\to\text{pt}_{\mathcal{C}}$ is $n$ -connected.

[05YI]

Lemma 4.2.5. Let $\mathcal{C}$ and $\mathcal{D}$ be $\infty$ -categories that admit finite limits and let $F\colon\mathcal{C}\leftrightarrows\mathcal{D}\penalty\mskip 6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0muG$ be an adjunction with $F\dashv G$ ,

(1)

For every $d\geq-2$ and a $d$ -truncated morphism $g$ in $\mathcal{D}$ , the morphism $G\left(g\right)$ is a $d$ -truncated morphism in $\mathcal{C}$ .
(2)

For every $n\geq-2$ and an $n$ -connected morphism $f$ in $\mathcal{C}$ , the morphism $F\left(f\right)$ is an $n$ -connected morphism in $\mathcal{D}$ .

[05YJ]

Proof. As a right adjoint, $G$ is left exact and therefore preserves $d$ -truncated morphisms by T.5.5.6.16. Since $G$ preserves $n$ -truncated morphisms and the space of lifts in the square

is homotopy equivalent to the space of lifts in the adjoint square

given by 4.1.4, we see that if $f$ is left orthogonal to all $n$ -truncated morphisms then so is $F\left(f\right)$ . ∎

[05YK]

Lemma 4.2.6. Let $\mathcal{C}$ be a presentable $\infty$ -category, let $f\colon A\to B$ be a morphism in $\mathcal{C}$ , and let $n\geq-2$ be an integer. The map $f$ is $n$ -connected if and only if viewed as an object $\overline{A}$ of $\mathcal{C}_{/B}$ , its $n$ -truncation $\tau_{\leq n}^{\mathcal{C}_{/B}}\left(\overline{A}\right)$ is the terminal object (ie $\operatorname{Id}_{B}\colon B\to B$ ).

[05YL]

Proof. Since $\mathcal{C}$ has all pullbacks, every commutative square $q\colon\Delta^{1}\times\Delta^{1}\to\mathcal{C}$ of the form

can be factored as

By 4.1.3, the space of lifts for the original square $q$ is equivalent to the space of lifts in the left square of the above rectangle. Moreover, $n$ -truncated morphisms are closed under base change and so to check that $f$ is $n$ -connected, we can equivalently restrict ourselves to checking the left orthogonality condition only for squares $q$ in which the map $B\to Y$ is the identity on $B$ . Writing $\overline{A}$ , $\overline{X}$ and $\overline{B}$ for $A\to B$ , $X\to B$ and $\operatorname{Id}\colon B\to B$ as objects of $\mathcal{C}_{/B}$ , respectively, we see that by the dual of T.5.5.5.12 the space of lifts fits into a fiber sequence

L\left(q\right)=\operatorname{Map}_{\mathcal{C}_{A//B}}\left(\overline{B},\overline{X}\right)\to\operatorname{Map}_{\mathcal{C}_{/B}}\left(\overline{B},\overline{X}\right)\xrightarrow{f^{*}}\operatorname{Map}_{\mathcal{C}_{/B}}\left(\overline{A},\overline{X}\right).

Hence, $f$ is $n$ -connected if and only if $f^{*}$ is an equivalence for every $n$ -truncated morphism $X\to B$ . By T.5.5.6.10, a morphism $X\to B$ is $n$ -truncated if and only if $\overline{X}$ is an $n$ -truncated object of $\mathcal{C}_{/B}$ . Hence, we need the above map to be an equivalence for every $n$ -truncated object $\overline{X}\in\mathcal{C}_{/B}$ . This precisely means that the map $\overline{A}\to\overline{B}$ exhibits $\overline{B}$ , the terminal object of $\mathcal{C}_{/B}$ , as the $n$ -truncation of $\overline{A}$ . ∎

[05YM]

Corollary 4.2.7. In a presentable $\infty$ -category $\mathcal{C}$ , an object $X$ is $n$ -connected for some $n\geq-2$ if and only if its $n$ -truncation $\tau_{\leq n}^{\mathcal{C}}X$ is a terminal object of $\mathcal{C}$ .

The following is a quantitative generalization of the defining property of an $n$ -connected morphism.

[05YN]

Proposition 4.2.8. Let $\mathcal{C}$ be a presentable $\infty$ -category. Fix integers $d\geq n\geq-2$ . For every square $q\colon\Delta^{1}\times\Delta^{1}\to\mathcal{C}$ of the form

in which $f\colon A\to B$ is $n$ -connected and $g\colon X\to Y$ is $d$ -truncated, the space of lifts $L\left(q\right)$ is $\left(d-n-2\right)$ -truncated.

[05YP]

Proof. We prove this by induction on $d$ . For $d=n$ , the claim follows from the definition of an $n$ -connected morphism and the fact that a space is $\left(-2\right)$ -connected if and only if it is contractible. We now assume that this is true for $d-1$ , and prove it for $d$ . Denote the space of lifts by $L\left(q\right)$ . By T.5.5.6.15, it suffices to show that the diagonal map $\delta\colon L\left(q\right)\to L\left(q\right)\times L\left(q\right)$ is $\left(d-n-3\right)$ -truncated. By 4.1.5, the homotopy fiber over a point $\left(s_{0},s_{1}\right)\in L\left(q\right)\times L\left(q\right)$ is equivalent to the space of lifts in the square

where the bottom map is $\left(s_{0},s_{1}\right)$ . By T.5.5.6.15, since $X\to Y$ is $d$ -truncated, $X\to X\times_{Y}X$ is $\left(d-1\right)$ -truncated and, therefore, by induction, the space of lifts is $\left(\left(d-1\right)-n-2\right)$ -truncated and we are done. ∎

Tomer Schlank, Lior Yanovski

Original source: arXiv:1808.06006v3

Original source · 1808.06006v3