The Injective category number on continuous maps

Abstract

We introduce the concept of injective category number $\text{IC}(f)$ for a continuous map $f\colon X\to~Y$, and present fundamental results concerning this numerical invariant. The value $\text{IC}(f)$ quantifies the \aspas{complexity} or \aspas{categorical structure} underlying the question: under what conditions is $f$ injective? More precisely, $\text{IC}(f)$ is the smallest positive integer $\ell$ such that $X$ can be covered by $\ell$ open subsets $U_1,\ldots,U_\ell$, with each restriction map $f_{\mid U}:U\to Y$ being injective. For instance, we examine the behaviour of $\text{IC}(f)$ under pullbacks and compositions of maps. In addition, we provide a cohomological lower bound for $\text{IC}(f)$. When $f$ has a finite number of multiple points, we express $\text{IC}(f)$ in terms of these points of non-injectivity. In the case that $f$ is the quotient map $\mathfrak{q}^X:X\to X/G$, where $X$ is a metric free $G$-space, we provide a lower bound for the injective category of $\mathfrak{q}^X$ in terms of the $2$-th index, $\text{ind}_2(X,G)$. When $G=\mathbb{Z}_2$, this lower bound is shown to be sharp. These results link a classical problem in Borsuk-Ulam theory to contemporary research developments in the study of injective category numbers.