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Visual difference between nominal and ordinal data (w/examples), the two scales of categorical data [2] A nominal variable, or nominal group, is a group of objects or ideas collectively grouped by a particular qualitative characteristic. [3] Nominal variables do not have a natural order, which means that statistical analyses of these variables ...
On the other hand, though the above properties guarantee the existence of a categorical equivalence (given a sufficiently strong version of the axiom of choice in the underlying set theory), the missing data is not completely specified, and often there are many choices. It is a good idea to specify the missing constructions explicitly whenever ...
A theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morley ( 1965 ) stating that if a first-order theory in a countable language is categorical in some uncountable cardinality , then it is categorical in all uncountable ...
Examples include Set and CPO, the category of complete partial orders with Scott-continuous functions. A topos is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets). A topos can also be used to represent a logical theory.
In a higher topos not only mathematics can be done but also "n-geometry", which is higher homotopy theory. The topos hypothesis is that the ( n +1)-category n Cat is a Grothendieck ( n +1)-topos. Higher topos theory can also be used in a purely algebro-geometric way to solve various moduli problems in this setting.
For this reason, it is used throughout mathematics. Applications to mathematical logic and semantics (categorical abstract machine) came later. Certain categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a foundation of mathematics. A topos can also be considered as a specific type of category ...
The categorical dual definition is a formal definition of a comonad (or cotriple); this can be said quickly in the terms that a comonad for a category is a monad for the opposite category. It is therefore a functor U {\displaystyle U} from C {\displaystyle C} to itself, with a set of axioms for counit and comultiplication that come from ...
Therefore, one can describe the state space of a gauge theory – or many kinds of quantum field theories (QFTs) and local quantum physics, in terms of the transformation groupoids given by symmetries, as for example in the case of a gauge theory, by the gauge transformations acting on states that are, in this case, connections.
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