ScalingStacks

Construction 2.21. Let $f\colon M\to N$ be a continuous map from a $B$-framed $n$-manifold to a $B^{\prime}$-framed $k$-manifold, possibly with boundary. Given a regularity condition on $f$, we will produce a composite map of colored operads

$$f^{-1}\colon\ddisk^{\partial,B^{\prime}}_{k/N}\longrightarrow\dmfld^{B}_{n/M}\longrightarrow\mfld^{B}_{n/M}~.$$

The second functor is the standard one. To describe the first functor we make use of Lemma 2.5 so that we can assume the maps $B\rightarrow\BTop(n)$ and $B^{\prime}\rightarrow\BTop(k)$ are equivalences. For this case, the first functor is given by $(U\hookrightarrow N)\mapsto(U\underset{N}{\times}M\hookrightarrow M)$, which is evidently functorial as well as monoidal.

Suppose the two restrictions

$$f_{|}\colon f^{-1}(N\smallsetminus\partial N)\to N\smallsetminus\partial N\qquad\text{ and }\qquad f_{|}\colon f^{-1}(\partial N)\to\partial N$$

are manifold bundles. Then, by inspection, this functor $f^{-1}$ carries isotopy equivalences to equivalences. Through Proposition 2.19, there results a multi-functor

(3)

$$f^{-1}\colon\disk^{B^{\prime}}_{k/N}\longrightarrow\mfld^{B}_{n/M}~.$$