971 research outputs found
On Khovanov's cobordism theory for su(3) knot homology
We reconsider the su(3) link homology theory defined by Khovanov in
math.QA/0304375 and generalized by Mackaay and Vaz in math.GT/0603307. With
some slight modifications, we describe the theory as a map from the planar
algebra of tangles to a planar algebra of (complexes of) `cobordisms with
seams' (actually, a `canopolis'), making it local in the sense of Bar-Natan's
local su(2) theory of math.GT/0410495.
We show that this `seamed cobordism canopolis' decategorifies to give
precisely what you'd both hope for and expect: Kuperberg's su(3) spider defined
in q-alg/9712003. We conjecture an answer to an even more interesting question
about the decategorification of the Karoubi envelope of our cobordism theory.
Finally, we describe how the theory is actually completely computable, and
give a detailed calculation of the su(3) homology of the (2,n) torus knots.Comment: 49 page
Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)
We give a scheme for interpreting shaded tangles as quantum programs, with
the property that isotopic tangles yield equivalent programs. We analyze many
known quantum programs in this way -- including entanglement manipulation and
error correction -- and in each case present a fully-topological formal
verification, yielding in several cases substantial new insight into how the
program works. We also use our methods to identify several new or generalized
procedures.Comment: In Proceedings QPL 2017, arXiv:1802.0973
Globular: an online proof assistant for higher-dimensional rewriting
This article introduces Globular, an online proof assistant for the
formalization and verification of proofs in higher-dimensional category theory.
The tool produces graphical visualizations of higher-dimensional proofs,
assists in their construction with a point-and- click interface, and performs
type checking to prevent incorrect rewrites. Hosted on the web, it has a low
barrier to use, and allows hyperlinking of formalized proofs directly from
research papers. It allows the formalization of proofs from logic, topology and
algebra which are not formalizable by other methods, and we give several
examples
Categorified cyclic operads
In this paper, we introduce a notion of categorified cyclic operad for
set-based cyclic operads with symmetries. Our categorification is obtained by
relaxing defining axioms of cyclic operads to isomorphisms and by formulating
coherence conditions for these isomorphisms. The coherence theorem that we
prove has the form "all diagrams of canonical isomorphisms commute". Our
coherence results come in two flavours, corresponding to the "entries-only" and
"exchangeable-output" definitions of cyclic operads. Our proof of coherence in
the entries-only style is of syntactic nature and relies on the coherence of
categorified non-symmetric operads established by Do\v{s}en and Petri\'c. We
obtain the coherence in the exchangeable-output style by "lifting" the
equivalence between entries-only and exchangeable-output cyclic operads, set up
by the second author. Finally, we show that a generalisation of the structure
of profunctors of B\' enabou provides an example of categorified cyclic operad,
and we exploit the coherence of categorified cyclic operads in proving that the
Feynman category for cyclic operads, due to Kaufmann and Ward, admits an odd
version.Comment: 57 page
Computing Critical Pairs in 2-Dimensional Rewriting Systems
International audienceRewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative for the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. Here, we are interested in proving confluence for polygraphs presenting 2-categories, which can be seen as a generalization of term rewriting systems. For this purpose, we propose an adaptation of the usual algorithm for computing critical pairs. Interestingly, this framework is much richer than term rewriting systems and requires the elaboration of a new theoretical framework for representing critical pairs, based on contexts in compact 2-categories
- …