Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An Introduction to Effectus Theory   Effectus theory is a new branch of categorical logic that aims to capture the
essentials of quantum logic, with probabilistic and Boolean logic as special
cases. Predicates in effectus theory are not subobjects having a Heyting
algebra structure, like in topos theory, but `characteristic' functions,
forming effect algebras. Such effect algebras are algebraic models of
quantitative logic, in which double negation holds. Effects in quantum theory
and fuzzy predicates in probability theory form examples of effect algebras.
  This text is an account of the basics of effectus theory. It includes the
fundamental duality between states and effects, with the associated Born rule
for validity of an effect (predicate) in a particular state. A basic result
says that effectuses can be described equivalently in both `total' and
`partial' form. So-called `commutative' and `Boolean' effectuses are
distinguished, for probabilistic and classical models. It is shown how these
Boolean effectuses are essentially extensive categories. A large part of the
theory is devoted to the logical notions of comprehension and quotient, which
are described abstractly as right adjoint to truth, and as left adjoint to
falisity, respectively. It is illustrated how comprehension and quotients are
closely related to measurement. The paper closes with a section on
`non-commutative' effectus theory, where the appropriate formalisation is not
entirely clear yet.

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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
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The categories of this paper are: stat
5
The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin