The Euler Characteristic of Finite Categories

We first assign a quadratic form and in particular a rational number to every finite category. In some cases, we call this rational number the Euler characteristic of the category. We show that this extends the definition of Leinster to a larger family of finite categories. Leinster's formula for the Euler characteristic of the Grothendieck construction of a diagram of categories requires the existence of a coweighting associated with the Grothendieck construction. Here we give a formula with no assumptions about the Grothedieck construction itself but only some conditions on the diagram. We also discuss the invariance and noninvariance of several definitions, in the literature, of Euler characteristics under weak equivalences of the canonical model category structure and the Thomason model category structure on categories respectively

Minimal models of some differential graded modules

Let $k$ be an algebraically closed field of characteristic $2$. Carlsson showed the existence of minimal models of certain semifree differential graded $k[x_1,\ldots,x_r]$-modules. Here we give an explicit construction of these minimal models without the semifreeness condition. We also prove Carlsson's rank conjecture when the degrees of all nonzero homology groups of a differential graded module have the same parity. Dually, we construct minimal models of chain complexes of Borel constructions of spaces with a free $(\mathbb{Z}/2)^r$-action. These minimal models are called minimal Hirsch-Brown models by Allday-Puppe. Puppe also showed that putting certain multiplicative structures on these minimal models could be useful to improve known results on related conjectures. Here we give our construction of these models using operadic language. This enables us to put multiplicative structures on these models.Comment: 8 page