995 research outputs found

### Rule Algebras for Adhesive Categories

We demonstrate that the most well-known approach to rewriting graphical
structures, the Double-Pushout (DPO) approach, possesses a notion of sequential
compositions of rules along an overlap that is associative in a natural sense.
Notably, our results hold in the general setting of $\mathcal{M}$-adhesive
categories. This observation complements the classical Concurrency Theorem of
DPO rewriting. We then proceed to define rule algebras in both settings, where
the most general categories permissible are the finitary (or finitary
restrictions of) $\mathcal{M}$-adhesive categories with $\mathcal{M}$-effective
unions. If in addition a given such category possess an $\mathcal{M}$-initial
object, the resulting rule algebra is unital (in addition to being
associative). We demonstrate that in this setting a canonical representation of
the rule algebras is obtainable, which opens the possibility of applying the
concept to define and compute the evolution of statistical moments of
observables in stochastic DPO rewriting systems

### Holography as a highly efficient RG flow II: An explicit construction

We complete the reformulation of the holographic correspondence as a
\emph{highly efficient RG flow} that can also determine the UV data in the
field theory in the strong coupling and large $N$ limit. We introduce a special
way to define operators at any given scale in terms of appropriate
coarse-grained collective variables, without requiring the use of the
elementary fields. The Wilsonian construction is generalised by promoting the
cut-off to a functional of these collective variables. We impose three criteria
to determine the coarse-graining. The first criterion is that the effective
Ward identities for local conservation of energy, momentum, etc. should
preserve their standard forms, but in new scale-dependent background metric and
sources which are functionals of the effective single trace operators. The
second criterion is that the scale-evolution equations of the operators in the
actual background metric should be state-independent, implying that the
collective variables should not explicitly appear in them. The final criterion
is that the endpoint of the scale-evolution of the RG flow can be transformed
to a fixed point corresponding to familiar non-relativistic equations with a
finite number of parameters, such as incompressible non-relativistic
Navier-Stokes, under a certain universal rescaling of the scale and of the time
coordinate. Using previous work, we explicitly show that in the hydrodynamic
limit each such highly efficient RG flow reproduces a unique classical gravity
theory with precise UV data that satisfy our IR criterion. We obtain the
explicit coarse-graining which reproduces Einstein's equations. In a simple
example, we are also able to compute the beta function. Finally, we show how
our construction can be interpolated with the traditional Wilsonian RG flow at
a suitable scale, and can be used to develop new non-perturbative frameworks
for QCD-like theories.Comment: 1+59 pages; Introduction slightly expanded, Section V on beta
function in highly efficient RG flow added, version accepted in PR

### Convolution Products on Double Categories and Categorification of Rule Algebras

Motivated by compositional categorical rewriting theory, we introduce a convolution product over presheaves of double categories which generalizes the usual Day tensor product of presheaves of monoidal categories. One interesting aspect of the construction is that this convolution product is in general only oplax associative. For that reason, we identify several classes of double categories for which the convolution product is not just oplax associative, but fully associative. This includes in particular framed bicategories on the one hand, and double categories of compositional rewriting theories on the other. For the latter, we establish a formula which justifies the view that the convolution product categorifies the rule algebra product

### Tracelet Hopf Algebras and Decomposition Spaces (Extended Abstract)

Tracelets are the intrinsic carriers of causal information in categorical
rewriting systems. In this work, we assemble tracelets into a symmetric
monoidal decomposition space, inducing a cocommutative Hopf algebra of
tracelets. This Hopf algebra captures important combinatorial and algebraic
aspects of rewriting theory, and is motivated by applications of its
representation theory to stochastic rewriting systems such as chemical reaction
networks.Comment: In Proceedings ACT 2021, arXiv:2211.0110

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