27,442 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
On the axiomatization of convex subsets of Banach spaces
We prove that any convex-like structure in the sense of Nate Brown is
affinely and isometrically isomorphic to a closed convex subset of a Banach
space. This answers an open question of Brown. As an intermediate step, we
identify Brown's algebraic axioms as equivalent to certain well-known axioms of
abstract convexity. We conclude with a new characterization of convex subsets
of Banach spaces.Comment: 8 pages, 1 figure. v3: added post-publication note on missing
reference with partly overlapping materia
Fractional Branes on a Non-compact Orbifold
Fractional branes on the non-compact orbifold \C^3/\Z_5 are studied. First,
the boundary state description of the fractional branes are obtained. The
open-string Witten index calculated using these states reproduces the adjacency
matrix of the quiver of . Then, using the toric crepant resolution of the
orbifold \C^3/\Z_5 and invoking the local mirror principle, B-type branes
wrapped on the holomorphic cycles of the resolution are studied. The boundary
states corresponding to the five fractional branes are identified as bound
states of BPS D-branes wrapping the 0-, 2- and 4-cycles in the exceptional
divisor of the resolution of \C^3/\Z_5.Comment: Latex2e, 25 pages, typos corrected, minor modifications, version to
appear in JHE
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