A pullback diagram in the coarse category (Q2680386)

The arena of coarse geometry investigates metric spaces as geometric objects from an asymptotic standpoint, boasting of Gromov's result on groups of polynomial growth [\textit{M. Gromov}, Publ. Math., Inst. Hautes Étud. Sci. 53, 53--78 (1981; Zbl 0474.20018)] and Yu's result on the Novikov conjecture [\textit{G. Yu}, Ann. Math. (2) 147, No. 2, 325--355 (1998; Zbl 0911.19001)] as key results. Although coarse geometry is not concrete, finite coproducts exist in the coarse category [\textit{E. Hartmann}, Math. Slovaca 70, No. 6, 1413--1444 (2020; Zbl 1505.51003)], being characterized as coarse covers in which every two elements are coarsely disjoint. Converely, products do not exist. This paper introduces a pullback diagram \[ \begin{array} [c]{ccl} & & Y\\ & & \downarrow d\left( \cdot,y_{0}\right) \\ X & \underset{d\left( \cdot,x_{0}\right) }{\rightarrow} & \mathbb{R}_{\geq0} \end{array} \] in the coarse category with fixed points \(x_{0}\in X\)\ and \(y_{0}\in Y\), the limit of which is called the \textit{asymptotic product}, denoted by \(X\ast Y\). Theorem 14 shows that it exists providing that the spaces \(X\)\ and \(Y\)\ are nice enough. If \(X\)\ and \(Y\)\ are hyperbolic proper geodestic metric spaces, then \(X\ast Y\)\ is hyperbolic geodestic proper [\textit{T. Foertsch} and \textit{V. Schroeder}, Geom. Dedicata 102, 197--212 (2003; Zbl 1048.53027)] so that its Gromov boundary \(\partial\left( X\ast Y\right) \)\ is defined, there existing a homeomorphism \[ \partial\left( X\ast Y\right) \cong\partial\left( X\right) \times \partial\left( Y\right) \] The above product construction allows of defining a homotopy theory on coarse metric spaces naturally. With \[ I=\mathbb{R}_{\geq0}\times\mathbb{R}_{\geq0} \] a coarse homotopy is defined to be a coarse map \[ H:X\ast I\rightarrow Y \] Theorem 28 shows that a flasque metric space \(X\) is coarssly homotopy equivalent to the product \(X\times\mathbb{Z}_{\geq0}\). \textit{M. Gromov} [Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. Proceedings of the symposium held at the Sussex University, Brighton, July 14-19, 1991. Cambridge: Cambridge University Press (1993; Zbl 0841.20039)] proposed the Lipshitz homotopy, which was exploited by \textit{J. Roe} [Index theory, coarse geometry, and topology of manifolds. Providence, RI: AMS, American Mathematical Society (1996; Zbl 0853.58003)] to show that controlled operator \(K\)-theory is a Lipshitz homotopy invariant on the one hand, and on the other hand by \textit{G. Yu} [Invent. Math. 139, No. 1, 201--240 (2000; Zbl 0956.19004)] to establish the coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. The homotopy theories in [\textit{P. D. Mitchener} et al., Math. Nachr. 293, No. 8, 1515--1533 (2020; Zbl 07261803)] is shown to be equivalent to the author's homotopy theory if the parameter function is chosen appropriately. The homotopy theory in turn is equivalent to the homotopy theory in [\textit{U. Bunke} and \textit{A. Engel}, Homotopy theory with bornological coarse spaces. Cham: Springer (2020; Zbl 1457.19001)] if the two parameter functions are chosen appropriately. The last section is concerned with colimits in the coarse category. It is shown that the coequalizer of two coarse maps with common domain and codomain exists, providing a proof that finite colimits exist in the coarse category.