Duality and traces for indexed monoidal categories (Q2847338)

It is well known [\textit{A. Dold} and \textit{D. Puppe}, in: Geometric topology, Proc. int. Conf., Warszawa 1978, 81--102 (1980; Zbl 0473.55008); Proc. Steklov Inst. Math. 154, 85--103 (1985; Zbl 0556.55006); \textit{A. Joyal} et al., Math. Proc. Camb. Philos. Soc. 119, No. 3, 447--468 (1996; Zbl 0845.18005); \textit{G. M. Kelly} and \textit{M. L. Laplaza}, J. Pure Appl. Algebra 19, 193--213 (1980; Zbl 0447.18005)] that in any symmetric monoidal category, there are useful intrinsic notions of \textit{duality} and \textit{trace}. NEWLINENEWLINEThis paper presents an abstract framework for constructing traces in \textit{indexed} symmetric monoidal categories, which gives rise to the Reidenmeister trace as a particular example. An indexed monoidal category is a family of symmetric monoidal category \(\mathcal{C}^{A}\), one for each object \(A\) of a cartesian monoidal base category \(\mathbf{S}\), equipped with base change functors indexed by the morphisms of \(\mathbf{S}\). The following three are primary examples in this paper.NEWLINENEWLINE\begin{itemize}NEWLINE\item[1.] \(\mathbf{S}=\) sets, \(\mathcal{C}^{A}=A\)-indexed families of abelian groups.NEWLINE\item[2.] \(\mathbf{S}=\) topological spaces, \(\mathcal{C}^{A}=\) spectra parametrized over \(A\).NEWLINE\item[3.] \(\mathbf{S}=\) sets, \(\mathcal{C}^{A}=A\)-indexed families of abelian groups.NEWLINE\end{itemize}NEWLINENEWLINEIn any such context one can consider duality and trace in the individual symmetric monoidal category \(\mathcal{C}^{A}\), i.e., \textit{fiberwise} [\textit{J. P. May} and \textit{J. Sigurdsson}, Parametrized homotopy theory. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1119.55001)]. The first main result in this paper, which is stated in \S 6 and is established in \S 11, goes as follows.NEWLINENEWLINETheorem 1. If \(M\in\mathcal{C}^{A}\) is a fiberwise dualizable and \(f:M\rightarrow M\) is any endomorphism, then the symmetric monoidal trace of \(f\) factors as a compositeNEWLINE\[NEWLINEI_{A}\rightarrow(\pi_{A})^{\ast}\langle \langle A\rangle \rangle \xrightarrow{\mathrm{tr}(\widehat{f})} I_{A}NEWLINE\]NEWLINENEWLINETrace-like information such as fixed point invariants can be extracted from an endomorphism \(f:M\rightarrow M\) in some cartesian monoidal category \(\mathbf{S}\) such as sets, groupoids or spaces, where a non-cartesian monoidal category \(\mathbf{C}\) such as abelian groups, chain complexes or spectra can be chosen, a functor \(\Sigma:\mathbf{S}\rightarrow \mathbf{C}\) such as the free abelian group or suspension spectrum is applied, and the symmetric monoidal trace \(\Sigma(f)\) in \(\mathbf{C}\) is considered. In most examples where this is done, there is actually an indexed symmetric monoidal category over \(\mathbf{S}\) such that \(\mathbf{C}=\mathcal{C}^{\ast}\) is the category indexed by the terminal object of \(\mathbf{S}\), and \(\Sigma(A)\) is the pushforward to \(\mathcal{C}^{\ast}\) of the unit object of \(\mathcal{C}^{A}\). The second main result in this paper, which is established in \S 8, goes as follows.NEWLINENEWLINETheorem 2. If \(I_{A}\) is totally dualizable [\textit{S. R. Costenoble} and \textit{S. Waner}, Equivariant ordinary homology and cohomology. Cham: Springer (2016; Zbl 1362.55001); \textit{J. P. May} and \textit{J. Sigurdsson}, Parametrized homotopy theory. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1119.55001)] and \(f:M\rightarrow M\) is an endomorphism in \(\mathbf{S}\), then the symmetric monoidal trace of \(\Sigma(f)\) factors as a compositeNEWLINE\[NEWLINEI_{\ast}\xrightarrow{\mathrm{tr}(\check{f})} \langle \langle A_{f}\rangle \rangle \rightarrow I_{\ast}NEWLINE\]NEWLINENEWLINEThe \textit{transfer} of \(f\), which is a map \(I_{\ast}\rightarrow \Sigma(A)\), is the trace of the compositeNEWLINE\[NEWLINE\Sigma(A)\xrightarrow{\Sigma(f)} \Sigma(A)\xrightarrow{\Sigma(\Delta_{A})} \Sigma(A)\otimes\Sigma(A)NEWLINE\]NEWLINEThe third main result in this paper, which is established in \S 8, goes as follows.NEWLINENEWLINETheorem 3. In the situation of the above theorem, the transfer of \(f\) factors as a compositeNEWLINE\[NEWLINEI_{\ast}\xrightarrow{\mathrm{tr}(\check{f})} \langle \langle A_{f}\rangle \rangle \rightarrow \Sigma(A)NEWLINE\]NEWLINENEWLINE\S \S 9--10 are devoted to string diagram calculus for indexed monoidal categories, which is a Poincaré dual way of drawing composition in categorical structures making complicated computations much more visually evident. String diagrams for monoidal categories and bicategories are described in [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 88, No. 1, 55--112 (1991; Zbl 0738.18005); \textit{A. Joyal} et al., Math. Proc. Camb. Philos. Soc. 119, No. 3, 447--468 (1996; Zbl 0845.18005); \textit{R. Penrose}, in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502); \textit{P. Selinger}, Lect. Notes Phys. 813, 289--355 (2011; Zbl 1217.18002); \textit{C. Heunen} and \textit{J. Vicary}, Categories for quantum theory. An introduction. Oxford: Oxford University Press (2019; Zbl 1436.81004); \textit{B. Coecke} and \textit{A. Kissinger}, Picturing quantum processes. A first course in quantum theory and diagrammatic reasoning. Cambridge: Cambridge University Press (2017; Zbl 1405.81001)].