Equivalenace of The Little Disk and Cacti operads

The subject of this thesis is to study the Little Disk operad and the Cacti operad and show that they are equivalent as operads as presented by Kaufmann in the article [Kau05]. In doing so, we go through a preliminary study of operads, what it means for them to be equivalent and the problems involved. We introduce and use the Little Disk in the process. We furthermore introduce and show results about the recognition principle of Fiedorowicz that is used to compare operads up against the Little Cube operad via a “ziq-zaq” through B∞ operads. We introduce and study the Cacti operad in detail while providing the means to finally apply the recognition principle. Throughout the thesis we will be elaborate on the graphical structures involved. This is both to fertilize the understanding of, but also to embrace the mathematical ideas and metaphors in, the subject. Resumé Emnet for nærværende speciale er at undersøge Lille Disk operaden og Kaktus operaden samt vise, at de er ækvivalente som operader, som det er præsenteret af Kaufmann i artiklen [Kau05]. I den forbindelse gennemg̊ar vi indledende studier af operader, hvad det vil sige, at operader er ækvivalente, og de problemer, der er involveret i det. Vi introducerer og anvender Lille Disk operaden i den proces. Derudover introducerer vi og viser resultater om “genkendelsesprincipet” af Feidorowicz, som der anvendes til at m̊ale operader om imod Lille Kube operaden via en “ziq-zaq” gennem B∞ operader. Vi introducerer Kaktus operaden i detaljer, mens vi fremstiller de betingelser, vi har brug for, for endeligt at kunne anvende “genkendelsesprincippet”. Gennem hele specialet er der fokus p̊a den grafiske fremstilling af emnet. Det er for b̊ade at understøtte forst̊aelsen af, men ogs̊a for at hylde de matematiske ideer og metaforer iboende i emnet.