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effectus

Effectus is a categorical framework for modeling and reasoning about effects that arise in computation and logic, including probabilistic, nondeterministic, and quantum phenomena. Introduced in the work of Bart Jacobs and collaborators, effectus theory aims to provide a unified language for states, predicates, and measurements across classical and quantum-like models, enabling compositional reasoning about complex systems.

In an effectus, objects represent systems and morphisms represent state-transforming processes. For each object X, there

A central feature of effectus theory is the state-effect duality. States are morphisms 1 → X from

Examples of effectuses include the category of sets with total functions (classical deterministic), the Kleisli category

is
a
distinguished
set
of
predicates
Pred(X),
consisting
of
arrows
X
→
2,
where
2
is
a
designated
truth-object.
Pred(X)
carries
the
structure
of
an
effect
algebra,
capturing
partial
addition
of
disjoint
predicates
and
a
notion
of
complement.
This
provides
a
generalization
of
the
familiar
Boolean
logic
to
systems
where
truth
values
may
be
probabilistic
or
quantum
in
nature.
The
theory
also
includes
a
notion
of
tests
and
assertions
that
generalize
measurements,
interacting
with
the
categorical
structure
(such
as
finite
coproducts)
in
precise
ways
to
support
a
compositional
semantics.
a
distinguished
unit
object
1,
while
effects
(predicates)
are
morphisms
X
→
2.
The
two
notions
are
connected
by
evaluation
maps
that
relate
how
a
given
state
satisfies
a
particular
predicate.
This
duality
underpins
a
unified
treatment
of
both
statistical
and
logical
aspects
of
a
system.
of
the
distribution
monad
(classical
probabilistic),
and
the
opposite
of
the
category
of
C*-algebras
with
normal
completely
positive
subunital
maps
(quantum).
Effectus
theory
also
connects
to
related
structures
such
as
effect
algebras
and
predicate-transformer
semantics,
informing
the
study
of
quantum
programming
languages
and
logical
systems.