Arboreal Categories: An Axiomatic Theory of Resources
Game comonads provide a categorical syntax-free approach to finite model theory, and their Eilenberg-Moore coalgebras typically encode important combinatorial parameters of structures. In this paper, we develop a framework whereby the essential properties of these categories of coalgebras are captur...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Logical Methods in Computer Science e.V.
2023-08-01
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| Series: | Logical Methods in Computer Science |
| Subjects: | |
| Online Access: | https://lmcs.episciences.org/9839/pdf |
| _version_ | 1797268517072928768 |
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| author | Samson Abramsky Luca Reggio |
| author_facet | Samson Abramsky Luca Reggio |
| author_sort | Samson Abramsky |
| collection | DOAJ |
| description | Game comonads provide a categorical syntax-free approach to finite model
theory, and their Eilenberg-Moore coalgebras typically encode important
combinatorial parameters of structures. In this paper, we develop a framework
whereby the essential properties of these categories of coalgebras are captured
in a purely axiomatic fashion. To this end, 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 general, axiomatic setting, and
applied to relational structures, where the comonadic constructions for
pebbling, Ehrenfeucht-Fra\"iss\'e and modal bisimulation games recently
introduced by Abramsky et al. are recovered, showing that many of the
fundamental notions of finite model theory and descriptive complexity arise
from instances of arboreal covers. |
| first_indexed | 2024-04-25T01:33:44Z |
| format | Article |
| id | doaj.art-b8c1b105022b40a3aa9f29f6346507fb |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:33:44Z |
| publishDate | 2023-08-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-b8c1b105022b40a3aa9f29f6346507fb2024-03-08T10:42:53ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742023-08-01Volume 19, Issue 310.46298/lmcs-19(3:14)20239839Arboreal Categories: An Axiomatic Theory of ResourcesSamson AbramskyLuca ReggioGame comonads provide a categorical syntax-free approach to finite model theory, and their Eilenberg-Moore coalgebras typically encode important combinatorial parameters of structures. In this paper, we develop a framework whereby the essential properties of these categories of coalgebras are captured in a purely axiomatic fashion. To this end, 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 general, axiomatic setting, and applied to relational structures, where the comonadic constructions for pebbling, Ehrenfeucht-Fra\"iss\'e and modal bisimulation games recently introduced by Abramsky et al. are recovered, showing that many of the fundamental notions of finite model theory and descriptive complexity arise from instances of arboreal covers.https://lmcs.episciences.org/9839/pdfcomputer science - logic in computer sciencemathematics - category theorymathematics - logic |
| spellingShingle | Samson Abramsky Luca Reggio Arboreal Categories: An Axiomatic Theory of Resources Logical Methods in Computer Science computer science - logic in computer science mathematics - category theory mathematics - logic |
| title | Arboreal Categories: An Axiomatic Theory of Resources |
| title_full | Arboreal Categories: An Axiomatic Theory of Resources |
| title_fullStr | Arboreal Categories: An Axiomatic Theory of Resources |
| title_full_unstemmed | Arboreal Categories: An Axiomatic Theory of Resources |
| title_short | Arboreal Categories: An Axiomatic Theory of Resources |
| title_sort | arboreal categories an axiomatic theory of resources |
| topic | computer science - logic in computer science mathematics - category theory mathematics - logic |
| url | https://lmcs.episciences.org/9839/pdf |
| work_keys_str_mv | AT samsonabramsky arborealcategoriesanaxiomatictheoryofresources AT lucareggio arborealcategoriesanaxiomatictheoryofresources |