Cohomological lower bound on the Lusternik–Schnirelmann category of a small category

To each topological space \(X\), one can assign the least cardinality of an open cover with each member contractible inside \(X\). If this cardinality is finite, then the related number is called the Lusternik–Schnirelmann category of \(X\). Recently, Kohei Tanaka modified the notion of Lusternik–Schnirelmann category (still a number) so that it is an invariant of (actual) small categories. A classical theorem says that the Lusternik–Schnirelmann category of a topological space can be bounded from below by its cup-length (a number determined by its singular cohomology ring). In the present paper, an analogous result is proved, where \(X\) is a small category. On the way to this result, an exposition of the nerve construction and cohomology of a small category is given, and the functoriality and “homotopy” invariance of this cohomology are proved.

Kohomološka spodnja meja za Lusternik–Schnirelmannovo število majhne kategorije

Za topološki prostor \(X\) se poraja vprašanje: kakšna je najmanjša kardinalnost odprtega pokritja \(X\), za katerega velja, da je vsaka odprta množica kontraktibilna znotraj \(X\)? Kadar je ta kardinalnost končna, se ustrezno število imenuje Lusternik–Schnirelmannova kategorija \(X\). Kohei Tanaka je pojem Lusternik–Schnirelmannove kategorije (število) priredil, tako da se aplicira na (dejanske) majhne kategorije. Klasični izrek pravi, da je Lusternik–Schnirelmannovi kategoriji topološkega prostora \(X\) mogoče postaviti spodnjo mejo – kupa-dolžino (angl. cup-length), definirano preko singularne kohomologije \(X\). V pričujočem članku je dokazan analogni izrek, kjer je \(X\) majhna kategorija. Na poti do tja so predstavljeni pojmi živca, kohomologije majhne kategorije, funktorialnosti in „homotopske“ invariantnosti te kohomološke konstrukcije.