1,009 research outputs found
Monads, partial evaluations, and rewriting
Monads can be interpreted as encoding formal expressions, or formal
operations in the sense of universal algebra. We give a construction which
formalizes the idea of "evaluating an expression partially": for example, "2+3"
can be obtained as a partial evaluation of "2+2+1". This construction can be
given for any monad, and it is linked to the famous bar construction, of which
it gives an operational interpretation: the bar construction induces a
simplicial set, and its 1-cells are partial evaluations.
We study the properties of partial evaluations for general monads. We prove
that whenever the monad is weakly cartesian, partial evaluations can be
composed via the usual Kan filler property of simplicial sets, of which we give
an interpretation in terms of substitution of terms.
In terms of rewritings, partial evaluations give an abstract reduction system
which is reflexive, confluent, and transitive whenever the monad is weakly
cartesian.
For the case of probability monads, partial evaluations correspond to what
probabilists call conditional expectation of random variables.
This manuscript is part of a work in progress on a general rewriting
interpretation of the bar construction.Comment: Originally written for the ACT Adjoint School 2019. To appear in
Proceedings of MFPS 202
Involutive Categories and Monoids, with a GNS-correspondence
This paper develops the basics of the theory of involutive categories and
shows that such categories provide the natural setting in which to describe
involutive monoids. It is shown how categories of Eilenberg-Moore algebras of
involutive monads are involutive, with conjugation for modules and vector
spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS)
construction is identified as a bijective correspondence between states on
involutive monoids and inner products. This correspondence exists in arbritrary
involutive categories
Topological Hochschild homology of Thom spectra and the free loop space
We describe the topological Hochschild homology of ring spectra that arise as
Thom spectra for loop maps f: X->BF, where BF denotes the classifying space for
stable spherical fibrations. To do this, we consider symmetric monoidal models
of the category of spaces over BF and corresponding strong symmetric monoidal
Thom spectrum functors. Our main result identifies the topological Hochschild
homology as the Thom spectrum of a certain stable bundle over the free loop
space L(BX). This leads to explicit calculations of the topological Hochschild
homology for a large class of ring spectra, including all of the classical
cobordism spectra MO, MSO, MU, etc., and the Eilenberg-Mac Lane spectra HZ/p
and HZ.Comment: 58 page
On coalgebras with internal moves
In the first part of the paper we recall the coalgebraic approach to handling
the so-called invisible transitions that appear in different state-based
systems semantics. We claim that these transitions are always part of the unit
of a certain monad. Hence, coalgebras with internal moves are exactly
coalgebras over a monadic type. The rest of the paper is devoted to supporting
our claim by studying two important behavioural equivalences for state-based
systems with internal moves, namely: weak bisimulation and trace semantics.
We continue our research on weak bisimulations for coalgebras over order
enriched monads. The key notions used in this paper and proposed by us in our
previous work are the notions of an order saturation monad and a saturator. A
saturator operator can be intuitively understood as a reflexive, transitive
closure operator. There are two approaches towards defining saturators for
coalgebras with internal moves. Here, we give necessary conditions for them to
yield the same notion of weak bisimulation.
Finally, we propose a definition of trace semantics for coalgebras with
silent moves via a uniform fixed point operator. We compare strong and weak
bisimilation together with trace semantics for coalgebras with internal steps.Comment: Article: 23 pages, Appendix: 3 page
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