Grothendieck groups and a categorification of additive invariants (Q2885375)

Given a homology theory, one can attempt to turn an additive homology class into a natural transformation. The authors suggest the following approach to a ``categorification'' of additive invariants. Consider a cospan of categories \(S: \mathcal C_s\to \mathcal B\) and \(T: \mathcal C_t\to \mathcal B\) where \(\mathcal B\) is a category with coproducts \(\sqcup\), \((\mathcal C_s, \sqcup)\) is a strict monoidal category with \(S\) a strict monoidal functor, and \(T\) is just a functor. (Here the notation \(\mathcal C_s\) and \(\mathcal C_t\) indicate source and target, respectively.) Choose an object \(X\) in \(\mathcal C_t\). Given two triples \((V,X,h)\) and \((V', X, h')\), \(V, V'\in \mathcal C_s\), \(h\in \hom_{\mathcal B}(S(V), T(X))\), \(h'\in \hom_{\mathcal B}(S(V'), T(X))\), we define them to be isomorphic if there exists an isomorphism \(\phi: V \to V'\) such that \(h=h'\circ S(\phi)\). The set of isomorphic classes can be turned into a monoid by the operation of disjoint union. Denote the associated Grothendieck group by \(K(S,T)(X)\), this is a covariant functor with respect to \(X\).NEWLINENEWLINEConsider a functor \(H: \mathcal B \to Ab\) where \(Ab\) is the category of abelian groups. Let \(\alpha\) be an additive invariant with values in \(H\), i.e. \(\alpha(V)\) is an object of \(H(\mathcal C_s(V))\) where \(V \in \mathcal C_s\). Given two triples \((V, X, h)\) and \((V', X, h')\) as above, we say that an isomorphism \(\phi: V \to V'\) is an \(\alpha\)-isomorphism if \(S(\phi)_*(\alpha(V))=\alpha(V')\). Now, given a class \([(V,X,h)]_{\alpha}\) of \(\alpha\)-isomorphisms, we turn the set of these classes to a monoid (using the additivity of \(\alpha\)). The corresponding Grothendieck group is denoted by \(K_{\alpha}(S,T)(X)\).NEWLINENEWLINETheorem (the categorification of an additive invariant) Let \(H:\mathcal B \to Ab\) be an additive functor on \(\mathcal B\) and \(T'=H\circ T\). Then an additive invariant \(\alpha\) as above induces a natural transformation on \(\mathcal C_t\): NEWLINE\[NEWLINE \tau_{\alpha}: K_{\alpha}(S,T)(-) \to T'(-), \quad \tau_{\alpha}([V,X,h]):=h_*(\alpha(V)). NEWLINE\]NEWLINENEWLINENEWLINEAs an example, let \(\mathcal B\) be the category \(TOP\) of topological spaces, \(T\) the identity functor id, and \(C\) the category of smooth closed manifolds with forgetful functor \(S: C\to \mathcal B\). Then we can construct additive invariants \(\alpha\) as follows. Let \([M]\) be the fundamental class of a smooth closed manifold \(M\). Given a characteristic class \(c: \text{Vect}(-) \to H^*(-;R)\), put \( \alpha([M])=c(TM)\cap [M]\in H_*(M;R). \) This gives us a natural transformation \(\tau_{\alpha}: K_{\alpha}(S, \text{id})(-) \to H_*(-;R)\). Now the oriented bordism group \(\Omega_*^{SO}(X)\) is isomorphic to \(K_{\alpha}(S, \text{id})(X)/\sim\) where \(\sim\) is the bordism relation. Furthermore, \(\tau_{\alpha}\) passes through \(\Omega_*^{SO}\) if \(c\) is a stable characteristic class.