Graph cohomology, colored posets and homological algebra in functor categories (Q2903922)

The author generalizes the homology theory of colored posets, defined by \textit{B. Everitt} and \textit{P. Turner} [J. Algebra 322, No. 2, 429--448 (2009; Zbl 1229.05293)]. In turn, Everett and Turners's theory is a generalization of homology theory of the \textit{M. Khovanov} cube construction used in [Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)] to define invariants of links.NEWLINENEWLINEA colored poset \((P,F)\) consists of a poset \(P\), with a unique maximal element 1, and a coloring \(F:P\rightarrow k\mathrm{-Mod}\), where \(k\) is a commutative ring and \(k\mathrm{-Mod}\) is the category of \(k\)-modules and \(k\)-homomorphisms. \(F\) is a functor on the category associated to \(P\) and denoted by the same letter. The homology groups of colored posets, \(\mathcal H (P,F)\), can be interpreted in terms of homological algebra in functor categories. For \(p\in P\), let \(k_p\) be an ``atomic'' functor on \(P\) such that \(k_p(p')=k\) if \(p=p'\), and \(k_p(p')=0\) otherwise. There exists an isomorphism \(\mathcal{H}_*(P,F)\cong \mathrm{Tor}^P_{\ast}(k_1,F)\). For a poset \(P\) with a unique minimal element \(o\) and a coloring \(F:P\rightarrow k\mathrm{-Mod}\) one can consider the cohomology groups \(\mathcal{H}^*(P,F)\cong \mathrm{Ext}^*_P(k_o,F)\). Let \(S[n]\) be the poset of all subsets of \([n]=\{0,1,\dots,n\}\) with inclusion as order relation. The poset \(S[n]\) has the minimal element (the empty set) \(o\) and the maximal element \([n]\). Together with a coefficient system \(F:S[n]\rightarrow k\mathrm{-Mod}\) it is an example of a colored poset. It is proved by Everitt and Turner [loc. cit.] that the groups \(\mathcal{H}_*(S[n],F)\) are isomorphic to the homology groups of the cube construction of a colored Boolean lattice. The cohomology groups of the Khovanov cube construction [loc. cit.] are isomorphic to \(H^*(S[n],F)\) for appropriate \(n\) and \(F\). If the edge set of a given graph \(\Gamma\) is equal to \([n]\), then the objects of \(S[n]\) can be considered as subgraphs of \(\Gamma\), with the same vertex set. For the Khovanov functor cohomology theory described by \textit{J. H. Przytycki} [Quantum Topol. 1, No. 2, 93--109 (2010; Zbl 1215.57006)] there is an isomorphism \(H^*(\Gamma,F)\cong Ext^*_{S[n]}(k_o,F)\cong \mathrm{Tor}^{S[n]}_{n+1-*}(k_{[n]},F)\). The cohomology theory of Przytycki is a generalization of the cohomology groups \(H^*(\Gamma,A)\), defined and studied by \textit{L. Helme-Guizon} and \textit{Y. W. Rong} [``Graph cohomologies from arbitrary algebras'', \url{arXiv:math/0506023}] and \textit{L. Helme-Guizon}, \textit{J. H. Przytycki} and \textit{Y. W. Rong} [Fundam. Math. 190, 139--177 (2006; Zbl 1105.57012)]. In this case the coeffients are given by an associative commutative unital \(k\)-algebra \(A\). It is proved by Przytycki [loc. cit.] that there exists a functor \(F_A:S[n]\rightarrow k\mathrm{-Mod}\), defined by \(A\), such that \(H^*(\Gamma,A)=H^*(\Gamma,F_A)\). Let \(\mathcal{F}\) be the category whose objects are sets \([n]\) , for all natural \(n\), and whose arrows are all mappings \(f:[n]\rightarrow [m]\). There is a functor \(C(A):\mathcal{F}\rightarrow k\mathrm{-Mod}\) [cf. \textit{J.-L. Loday}, Cyclic homology. Berlin: Springer-Verlag (1992; Zbl 0780.18009)] such that \(C(A)[n]=A^{\otimes n+1}\).NEWLINENEWLINEIn Section 2, the author defines a functor \(\pi_{\Gamma}:S[n]\rightarrow \mathcal{P}\), where \(\mathcal{P}\) is a certain subcategory of \(\mathcal{F}\) considered by \textit{M. Zimmermann} [Complexes de chaînes et petites catégories. Strasbourg: Univ. Louis Pasteur (Thèse) (2004)]. If a geometrical realization of \(\Gamma'\) has \(k+1\) connected components, then \(\pi_{\Gamma}(\Gamma')=[k]\). It follows from definition that \(f_A=C(A)\pi_{\Gamma}\) so that \(H^*(\Gamma,A)\cong \mathcal{H}^*(S[n],C(A)\pi_{\Gamma})\cong H^*(\Gamma,C(A)\;\pi_{\Gamma})\).NEWLINENEWLINEThe author considers the graph cohomology groups \(H^*(\Gamma,F\pi_{\Gamma})\), where \(F:\mathcal{P}\rightarrow k\mathrm{-Mod}\) is an arbitrary functor. Each graph \(\Gamma\) with \(n+1\) vertices defines a subcategory \(P(\Gamma)\) of the partition category \(P_n\), associated to the poset of all equivalence relations on the set \([n]\downarrow \mathcal{\mathcal{P}}\) whose objects are the arrows \(f:[n]\rightarrow [m]\) of \(\mathcal{P}\). There exist functors \(\rho_{\Gamma}:S[n]\rightarrow P(\Gamma)\) and \(\overline{\psi}:P_n\rightarrow \mathcal{P}\) such that \(\overline{\psi}\rho_{\Gamma}=\pi_{\Gamma}\). The author proves, that there are isomorphisms \(H^*(\Gamma,F\pi_{\Gamma})\cong \mathcal{H}(P(\Gamma),F\overline{\psi})\).NEWLINENEWLINEIn Section 1, the author considers, as a generalization of colored posets, small categories with objects for which atomic functors can be defined. As an example one can take \(\mathcal{P}\). If \(\Gamma\) is a full graph with \(n+1\) vertices, then \(P(\Gamma)=P_n\) and \(H^*(\Gamma,F\pi_{\Gamma})\cong \mathrm{Ext}^*_{\mathcal{P}}(k_{[n]},F)\).NEWLINENEWLINEFinally, some functorial properties of these cohomology theories are studied and the connection between the Hochschild homology of an algebra and the graph cohomology, defined for the same algebra and a cyclic graph, is explained by the author from the point of view of homological algebra in functor categories.