Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Homotopy of Walks and Spherical Maps in Homotopy Type Theory   We work with combinatorial maps to represent graph embeddings into surfaces
up to isotopy. The surface in which the graph is embedded is left implicit in
this approach. The constructions herein are proof-relevant and stated with a
subset of the language of homotopy type theory.
  This article presents a refinement of one characterisation of embeddings in
the sphere, called spherical maps, of connected and directed multigraphs with
discrete node sets. A combinatorial notion of homotopy for walks and the normal
form of walks under a reduction relation is introduced. The first
characterisation of spherical maps states that a graph can be embedded in the
sphere if any pair of walks with the same endpoints are merely walk-homotopic.
The refinement of this definition filters out any walk with inner cycles. As we
prove in one of the lemmas, if a spherical map is given for a graph with a
discrete node set, then any walk in the graph is merely walk-homotopic to a
normal form.
  The proof assistant Agda contributed to formalising the results recorded in
this article.

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The categories of this paper are: hep-ex
19
The categories of this paper are: eess
11
The categories of this paper are: hep-lat
37
The categories of this paper are: supr-con
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
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The categories of this paper are: astro-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat