Corpus ID: 231934179

Arboreal Categories: An Axiomatic Theory of Resources

@article{Abramsky2021ArborealCA,
  title={Arboreal Categories: An Axiomatic Theory of Resources},
  author={Samson Abramsky and Luca Reggio},
  journal={ArXiv},
  year={2021},
  volume={abs/2102.08109}
}
We introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of a game, to be defined. These are related to extensional or “static” structures via arboreal covers, which are resource-indexed comonadic adjunctions. These ideas are developed in a very general, axiomatic setting, and applied to relational structures, where the comonadic constructions for pebbling… Expand
Arboreal Categories and Resources
We introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of aExpand
The Pebble-Relation Comonad in Finite Model Theory
The pebbling comonad, introduced by Abramsky, Dawar and Wang, provides a categorical interpretation for the k-pebble games from finite model theory. The coKleisli category of the pebbling comonadExpand

References

SHOWING 1-10 OF 28 REFERENCES
Relating Structure and Power: Comonadic Semantics for Computational Resources
TLDR
The results pave the way for systematic connections between two major branches of the field of logic in computer science which hitherto have been almost disjoint: categorical semantics, and finite and algorithmic model theory. Expand
Comonadic semantics for guarded fragments
  • S. Abramsky, Dan Marsden
  • Computer Science, Mathematics
  • 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
  • 2021
TLDR
A systematic account of how a range of model comparison games which play a central role in finite model theory can be captured in terms of resource-indexed comonads on the category of relational structures, including Ehrenfeucht-Fraïssé, pebbling, and bisimulation games, is extended to quantifier-guarded fragments of first-order logic. Expand
Bisimulation and open maps
  • A. Joyal, M. Nielsen, G. Winskel
  • Mathematics, Computer Science
  • [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science
  • 1993
TLDR
An abstract definition of bisimulation is presented and a promising new model, presheaf on categories of pomsets, into which the usual category of labeled event structures embeds fully and faithfully is presented. Expand
Bisimulation from Open Maps
TLDR
A logic, generalising Hennessy?Milner logic, which is characteristic for the generalised notion of bisimulation is presented, which makes possible a uniform definition of bisIMulation across a range of different models for parallel computation presented as categories. Expand
Full Abstraction for PCF
An intensional model for the programming language PCF is described, in which the types of PCF are interpreted by games, and the terms by certain "history-free" strategies. This model is shown toExpand
On Full Abstraction for PCF: I, II, and III
TLDR
An order-extensional, order (or inequationally) fully abstract model for Scott's language pcf, based on a kind of game in which each play consists of a dialogue of questions and answers between two players who observe the following principles of civil conversation. Expand
Homomorphism preservation theorems
TLDR
It is proved that the h.p.t. remains valid when restricted to finite structures (unlike many other classical preservation theorems, including the Łoś--Tarski theorem and Lyndon's positivity theorem). Expand
∞-Categories for the Working Mathematician
homotopy theory C.1. Lifting properties, weak factorization systems, and Leibniz closure C.1.1. Lemma. Any class of maps characterized by a right lifting property is closed under composition,Expand
FACTORIZATION SYSTEMS
These notes were written to accompany a talk given in the Algebraic Topology and Category Theory Proseminar in Fall 2008 at the University of Chicago. We first introduce orthogonal factorizationExpand
Tree-depth, subgraph coloring and homomorphism bounds
TLDR
The notions tree-depth and upper chromatic number of a graph are defined and it is shown that the upper chromatics number coincides with the maximal function which can be locally demanded in a bounded coloring of any proper minor closed class of graphs. Expand
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