52 research outputs found
Brown's moduli spaces of curves and the gravity operad
This paper is built on the following observation: the purity of the mixed
Hodge structure on the cohomology of Brown's moduli spaces is essentially
equivalent to the freeness of the dihedral operad underlying the gravity
operad. We prove these two facts by relying on both the geometric and the
algebraic aspects of the problem: the complete geometric description of the
cohomology of Brown's moduli spaces and the coradical filtration of cofree
cooperads. This gives a conceptual proof of an identity of Bergstr\"om-Brown
which expresses the Betti numbers of Brown's moduli spaces via the inversion of
a generating series. This also generalizes the Salvatore-Tauraso theorem on the
nonsymmetric Lie operad.Comment: 26 pages; corrected Figure
New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic
Intuitionistic logic, in which the double negation law not-not-P = P fails,
is dominant in categorical logic, notably in topos theory. This paper follows a
different direction in which double negation does hold. The algebraic notions
of effect algebra/module that emerged in theoretical physics form the
cornerstone. It is shown that under mild conditions on a category, its maps of
the form X -> 1+1 carry such effect module structure, and can be used as
predicates. Predicates are identified in many different situations, and capture
for instance ordinary subsets, fuzzy predicates in a probabilistic setting,
idempotents in a ring, and effects (positive elements below the unit) in a
C*-algebra or Hilbert space. In quantum foundations the duality between states
and effects plays an important role. It appears here in the form of an
adjunction, where we use maps 1 -> X as states. For such a state s and a
predicate p, the validity probability s |= p is defined, as an abstract Born
rule. It captures many forms of (Boolean or probabilistic) validity known from
the literature. Measurement from quantum mechanics is formalised categorically
in terms of `instruments', using L\"uders rule in the quantum case. These
instruments are special maps associated with predicates (more generally, with
tests), which perform the act of measurement and may have a side-effect that
disturbs the system under observation. This abstract description of
side-effects is one of the main achievements of the current approach. It is
shown that in the special case of C*-algebras, side-effect appear exclusively
in the non-commutative case. Also, these instruments are used for test
operators in a dynamic logic that can be used for reasoning about quantum
programs/protocols. The paper describes four successive assumptions, towards a
categorical axiomatisation of quantitative logic for probabilistic and quantum
systems
Weakly Markov Categories and Weakly Affine Monads
Introduced in the 1990s in the context of the algebraic approach to graph rewriting, gs-monoidal categories are symmetric monoidal categories where each object is equipped with the structure of a commutative comonoid. They arise for example as Kleisli categories of commutative monads on cartesian categories, and as such they provide a general framework for effectful computation. Recently proposed in the context of categorical probability, Markov categories are gs-monoidal categories where the monoidal unit is also terminal, and they arise for example as Kleisli categories of commutative affine monads, where affine means that the monad preserves the monoidal unit.
The aim of this paper is to study a new condition on the gs-monoidal structure, resulting in the concept of weakly Markov categories, which is intermediate between gs-monoidal categories and Markov ones. In a weakly Markov category, the morphisms to the monoidal unit are not necessarily unique, but form a group. As we show, these categories exhibit a rich theory of conditional independence for morphisms, generalising the known theory for Markov categories. We also introduce the corresponding notion for commutative monads, which we call weakly affine, and for which we give two equivalent characterisations.
The paper argues that these monads are relevant to the study of categorical probability. A case at hand is the monad of finite non-zero measures, which is weakly affine but not affine. Such structures allow to investigate probability without normalisation within an elegant categorical framework
Weakly Markov categories and weakly affine monads
Introduced in the 1990s in the context of the algebraic approach to graph
rewriting, gs-monoidal categories are symmetric monoidal categories where each
object is equipped with the structure of a commutative comonoid. They arise for
example as Kleisli categories of commutative monads on cartesian categories,
and as such they provide a general framework for effectful computation.
Recently proposed in the context of categorical probability, Markov categories
are gs-monoidal categories where the monoidal unit is also terminal, and they
arise for example as Kleisli categories of commutative affine monads, where
affine means that the monad preserves the monoidal unit.
The aim of this paper is to study a new condition on the gs-monoidal
structure, resulting in the concept of weakly Markov categories, which is
intermediate between gs-monoidal categories and Markov ones. In a weakly Markov
category, the morphisms to the monoidal unit are not necessarily unique, but
form a group. As we show, these categories exhibit a rich theory of conditional
independence for morphisms, generalising the known theory for Markov
categories. We also introduce the corresponding notion for commutative monads,
which we call weakly affine, and for which we give two equivalent
characterisations.
The paper argues that these monads are relevant to the study of categorical
probability. A case at hand is the monad of finite non-zero measures, which is
weakly affine but not affine. Such structures allow to investigate probability
without normalisation within an elegant categorical framework.Comment: CALCO 202
Categories for Me, and You?
A non-self-contained gathering of notes on category theory, including the definition of locally cartesian closed category, of the cartesian structure in slice categories, or of the “pseudo-cartesian structure” on Eilenberg–Moore categories. References and proofs are provided, sometimes, to my knowledge, for the first time
Shades of Iteration: from Elgot to Kleene
Notions of iteration range from the arguably most general Elgot iteration to
a very specific Kleene iteration. The fundamental nature of Elgot iteration has
been extensively explored by Bloom and Esik in the form of iteration theories,
while Kleene iteration became extremely popular as an integral part of
(untyped) formalisms, such as automata theory, regular expressions and Kleene
algebra. Here, we establish a formal connection between Elgot iteration and
Kleene iteration in the form of Elgot monads and Kleene monads, respectively.
We also introduce a novel class of while-monads, which like Kleene monads admit
a relatively simple description in algebraic terms. Like Elgot monads,
while-monads cover a large variety of models that meaningfully support
while-loops, but may fail the Kleene algebra laws, or even fail to support a
Kleen iteration operator altogether.Comment: Extended version of the accepted one for "Recent Trends in Algebraic
Development Techniques - 26th IFIP WG 1.3 International Workshop, WADT 2022
A Fibrational Framework for Substructural and Modal Logics
We define a general framework that abstracts the common features of many intuitionistic substructural and modal logics / type theories. The framework is a sequent calculus / normal-form type theory parametrized by a mode theory, which is used to describe the structure of contexts and the structural properties they obey. In this sequent calculus, the context itself obeys standard structural properties, while a term, drawn from the mode theory, constrains how the context can be used. Product types, implications, and modalities are defined as instances of two general connectives, one positive and one negative, that manipulate these terms. Specific mode theories can express a range of substructural and modal connectives, including non-associative, ordered, linear, affine, relevant, and cartesian products and implications; monoidal and non-monoidal functors, (co)monads and adjunctions; n-linear variables; and bunched implications. We prove cut (and identity) admissibility independently of the mode theory, obtaining it for many different logics at once. Further, we give a general equational theory on derivations / terms that, in addition to the usual beta/eta-rules, characterizes when two derivations differ only by the placement of structural rules. Additionally, we give an equivalent semantic presentation of these ideas, in which a mode theory corresponds to a 2-dimensional cartesian multicategory, the framework corresponds to another such multicategory with a functor to the mode theory, and the logical connectives make this into a bifibration. Finally, we show how the framework can be used both to encode existing existing logics / type theories and to design new ones
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