Subsection 13.3. (0MTX) · ScalingStacks

13.3. Categorical equivalences[0MTX]

We say that a map $g\colon U\rightarrow V$ of Segal spaces is a categorical equivalence if there exist maps $f,h\colon V\rightarrow U$ and categorical homotopies $gf\sim 1_{V}$ and $hg\sim 1_{U}$ . Note that if $U$ and $V$ are discrete nerves of categories, then the categorical equivalences correspond exactly to equivalences of categories.

[05W6]

Proposition 13.4. A map $g\colon U\rightarrow V$ between complete Segal spaces is a categorical equivalence if and only if it is a simplicial homotopy equivalence, if and only if it is a Reedy weak equivalence.

[05W7]

Proof. The first “if and only if” is immediate from (13.2), while the second follows from the fact that complete Segal spaces are cofibrant and fibrant in the Reedy simplicial model category. ∎

[05W8]

Proposition 13.5. Let $A$ , $B$ , and $W$ be Segal spaces. If $f,g\colon A\rightrightarrows B$ are categorically homotopic maps, then the induced maps $W^{B}\rightrightarrows W^{A}$ are categorically homotopic. If $f\colon A\rightarrow B$ is a categorical equivalence, then the induced map $W^{B}\rightarrow W^{A}$ is a categorical equivalence.

[05W9]

Proof. If a categorical homotopy between $f$ and $g$ is given by $H\colon A\times E\rightarrow B$ , then $W^{H}\colon W^{B}\rightarrow W^{A\times E}\approx(W^{A})^{E}$ is a categorical homotopy of $W^{f}$ and $W^{g}$ . The statement about categorical equivalences follows. ∎

[05WA]

Proposition 13.6. If $f\colon U\rightarrow V$ is a categorical equivalence between Segal spaces, then it is a weak equivalence in the complete Segal space model category structure.

[05WB]

Proof. Recall from (7.2) that $f$ is a weak equivalence in the complete Segal space model category if and only if $\Map_{s{\operatorname{\mathcal{S}}}}(f,W)$ is a weak equivalence of spaces for each complete Segal space $W$ . This is equivalent to supposing that $W^{f}\colon W^{V}\rightarrow W^{U}$ is a Reedy weak equivalence for each complete Segal space $W$ , since $(W^{f})_{n}\approx\Map_{s{\operatorname{\mathcal{S}}}}(f,W^{F(n)})$ and since $W^{F(n)}$ is a complete Segal space by (7.3). The result now follows by noting that $W^{f}$ is a categorical equivalence between complete Segal spaces by (13.5) and (7.3), and thus is a Reedy weak equivalence by (13.4). ∎