You are cordially invited to the Ph.D. Thesis Defense Presentation<\/p>\n
\u201cThe Euler Measure of Finite Categories\u201d<\/p>\n
Mustafa Akkaya, Ph. D. Student in Mathematics<\/p>\n
Abstract: We associate a rational number $chi(mathcal{A})$ to every category $mathcal{A}$ whose object and morphism sets are finite. The assignment $chi $ is additive under disjoint union and it preserves products. Leinster\u2019s Euler characteristic $chi_{Lein}$ and $chi$ agrees whenever $chi_{Lein}$ is defined. Hence $chi$ is different from series Euler characteristic $chi_{Sigma}$ and $chi$ is preserved under the weak equivalences of canonical model structure when it is restricted to the family of categories for which $chi_{Lein}$ is defined. However, $chi$ is not preserved under the weak equivalences of canonical model structure on its whole domain. For this reason $chi$ is not called Euler characteristic. When the domain of $chi$ is restricted to the family of categories admitting a weighting, $chi$ satisfies the inclusion exclusion principle. Hence we can call this restriction the Euler measure. By abuse of notation we will denote this restriction by $chi$ again. Since the family of categories admitting both weighting and coweighting is contained by the family of categories admitting weighting, the Euler measure of categories is a proper extension of Leinster\u2019s Euler characteristic. We also showed that Leinster\u2019s formula for Grothendieck construction is still valid for diagrams from a poset to the categories in the domain this Euler measure. Hence this can be used to give results about homotopy colimits in canonical model structure. The situation for Thomason model structure is more intricate. We give an example to show that non of $chi$, $chi_{Lein}$ and $chi_{Sigma}$ is invariant under the weak equivalences of Thomason model structure and show that such examples can be eliminated by putting extra conditions on weak equivalences of Thomason model structure.<\/p>\n
Advisor: Assist. Prof. Dr. \u00d6zg\u00fcn \u00dcnl\u00fc
\nDate: 3 September 2024
\nTime: 13:00
\nPlace: Department of Mathematics Seminar Room SA-141<\/p>\n","protected":false},"excerpt":{"rendered":"
You are cordially invited to the Ph.D. Thesis Defense Presentation \u201cThe Euler Measure of Finite Categories\u201d Mustafa Akkaya, Ph. D. Student in Mathematics Abstract: We associate a rational number $chi(mathcal{A})$ to every category $mathcal{A}$ whose object and morphism sets are finite. The assignment $chi $ is additive under disjoint union [\u2026] <\/p>\n