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This paper deals with different concepts for characterizing the size ofmathematical objects. A game theoretic investigation and generalization of twosize concepts, which can both be formulated in topological terms, is provided:the so called "Baire category" and the "$\sigma $-category". This is mainlydone by means of (generalized) Banach-Mazur games using the Axiom ofdeterminacy (the inconsistency of AC and AD is reflected in the beginning and aweaker form of AC is chosen for proofs). Analogue versions of Cantor-Bendixsonand Heine-Borel are proofed as well as some definability results.  Such size concepts are established f.i. in measure and integration theory,set theory and topology often leading to mathematically precise formulations of"fuzziness". In measure and integration theory f.i. one defines for a "measurespace" $(\Omega, \underline{A}, \mu)$ - i.e. $\Omega$ is a non empty set,$\underline{A}$ a $\sigma$-algebra on $\Omega$ and $\mu$ a measure on$(\Omega,\underline{A})$ - and a property $E \subset \Omega$ of elements of$\Omega$, that the property $E$ is valid in a set $A\in\underline{A}$"$\mu$-almost everywhere", if $E$ is true in $A$ up to a "$\mu$-null set" -i.e. $\exists N\in\underline{A}{ } (\mu(N)=0 \wedge A\cap N^\complement \subsetE)$. This is crucial for the formulation of uniqueness statements in measureand integration theory as its theorems usually only apply up to "small"($\mu$-null) sets.  Key words: Axiom of choice AC, Axiom of determinacy AD, Baire categorytheorem, Baire property, Baire space, Banach-Mazur games, Borel hierachy,Cantor-Bendixson, definability, Gale-Stewart, Heine-Borel, Lusin hierarchy,meager sets, perfect set property, polish spaces, projective hierarchy, sigmabounded sets, sigma compact sets, superperfect sets, topological games, winningstrategy

Categorical concepts and their generalization by means of game theory (german: Kategorien-Konzepte und ihre spieltheoretische Verallgemeinerung)