Non-symmetric exponential laws in the construct \({\mathcal P}r{\mathcal T}op\) (Q1610280)

The categories \({\mathcal T}op\) of topological spaces and \({\mathcal P}r{\mathcal T}op\) of pretopological (closure) spaces are not Cartesian closed. If one weakens Cartesian closedness to the symmetric monoidal property, both \({\mathcal T}op\) and \({\mathcal P}r{\mathcal T}op\) have just one such structure [\textit{J. Činčura}, Math. Slovaca 29, 245-255 (1979; Zbl 0419.18001); Commentat. Math. Univ. Carol. 20, 431-446 (1979; Zbl 0416.18014)]. When removing the symmetry condition, there is a proper class of monoidal structures in \({\mathcal T}op\) [\textit{R. Brown}, Q. J. Math., Oxf. II. Ser. 15, 238-250 (1964; Zbl 0126.38503)]; the present paper proves the same result for the category \({\mathcal P}r{\mathcal T}op\).