Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Coarse-Graining Open Markov Processes   Coarse-graining is a standard method of extracting a simple Markov process
from a more complicated one by identifying states. Here we extend
coarse-graining to open Markov processes. An "open" Markov process is one where
probability can flow in or out of certain states called "inputs" and "outputs".
One can build up an ordinary Markov process from smaller open pieces in two
basic ways: composition, where we identify the outputs of one open Markov
process with the inputs of another, and tensoring, where we set two open Markov
processes side by side. In previous work, Fong, Pollard and the first author
showed that these constructions make open Markov processes into the morphisms
of a symmetric monoidal category. Here we go further by constructing a
symmetric monoidal double category where the 2-morphisms are ways of
coarse-graining open Markov processes. We also extend the already known
"black-boxing" functor from the category of open Markov processes to our double
category. Black-boxing sends any open Markov process to the linear relation
between input and output data that holds in steady states, including
nonequilibrium steady states where there is a nonzero flow of probability
through the process. To extend black-boxing to a functor between double
categories, we need to prove that black-boxing is compatible with
coarse-graining.

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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: q-alg
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The categories of this paper are: chem-ph
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The categories of this paper are: cs
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The categories of this paper are: solv-int
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The categories of this paper are: econ