Abstract
This paper considers the difficulty in the set-system approach to generalizing graph theory. These difficulties arise categorically as the category of set-system hypergraphs is shown not to be cartesian closed and lacks enough projective objects, unlike the category of directed multigraphs (i.e., quivers). The category of incidence hypergraphs is introduced as a “graph-like” remedy for the set-system issues so that hypergraphs may be studied by their locally graphic behavior via homomorphisms that allow an edge of the domain to be mapped into a subset of an edge in the codomain. Moreover, it is shown that the category of quivers embeds into the category of incidence hypergraphs via a logical functor that is the inverse image of an essential geometric morphism between the topoi. Consequently, the quiver exponential is shown to be simply represented using incidence hypergraph homomorphisms.
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References
Berge, C.: Hypergraphs, North-Holland Mathematical Library, vol. 45, North-Holland Publishing Co., Amsterdam, (1989). Combinatorics of finite sets, Translated from the French
Bondy, J.A., Murty, U.S.R.: Graph theory, Graduate Texts in Mathematics, vol. 244. Springer, New York (2008)
Borceux, F.: Handbook of categorical algebra. 1-3, Encyclopedia of Mathematics and its Applications, vol. 50-52, Cambridge University Press, Cambridge, (1994)
Brown, R., Morris, I., Shrimpton, J., Wensley, C.D.: Graphs of morphisms of graphs, electron. J. Combin. 15(1), 1–28 (2008)
Bumby, R.T., Latch, D.M.: Categorical constructions in graph theory. Int. J. Math. Math. Sci. 9(1), 1–16 (1986)
Chen, G., Liu, V., Robinson, E., Rusnak, L.J., Wang, K.: A characterization of oriented hypergraphic laplacian and adjacency matrix coefficients. Linear Algebra Appl. 556, 323–341 (2018)
Chen, V., Rao, A., Rusnak, L.J., Yang, A.: A characterization of oriented hypergraphic balance via signed weak walks. Linear Algebra Appl. 485, 442–453 (2015)
Diestel, Reinhard: Graph theory, Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2010)
Dörfler, W., Waller, D.A.: A category-theoretical approach to hypergraphs. Arch. Math. 34(2), 185–192 (1980)
Duchet, P.: Hypergraphs, Handbook of combinatorics, vol. 1, 2, Elsevier, Amsterdam, (1995)
El-Zahar, M., Sauer, N.: The chromatic number of the product of two 4-chromatic graphs is 4. Combinatorica 5(2), 121–126 (1985)
Grilliette, W.: Injective envelopes and projective covers of quivers. Electron. J. Combin. 19(2), 39 (2012)
Grilliette, W., Seacrest, D., Seacrest, T.: On blow-ups and injectivity of quivers. Electron. J. Combinatorics. 20(2), 40 (2013)
Herrlich, H., Strecker, G.E.: Abstract and concrete categories: the joy of cats. Repr. Theory Appl. Categ. vol. 17, Wiley, New York, (2006)
Hilton, P.J., Stammbach, U.: A course in homological algebra, Graduate Texts in Mathematics, vol. 4, Springer-Verlag, (1997)
Johnstone, PT.: Sketches of an elephant: a topos theory compendium. Vol. 1, Oxford Logic Guides, vol. 43, The Clarendon Press, Oxford University Press, New York, (2002)
Mac Lane, S., Moerdijk, I.: Sheaves in geometry and logic, Universitext, Springer-Verlag, New York, (1994), A first introduction to topos theory, Corrected reprint of the 1992 edition
Mac Lane, S.: Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, Springer-Verlag, (1998)
Montanari, U., Ehrig, H., Kreowski, H.J., Rozenberg, G. (eds.): Handbook of graph grammars and computing by graph transformation. Vol. 3. Concurrency, parallelism, and distribution, World Scientific Publishing Co., Inc., River Edge, NJ, (1999)
Prange, U., Ehrig, H., Ehrig, K., Taentzer, G.: Fundamentals of algebraic graph transformation. Monographs in Theoretical Computer Science, Springer-Verlag, (2006)
Pavel, H., Jaroslav, N.: Graphs and homomorphisms, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, (2004)
Raeburn, I.: Graph algebras, CBMS Regional Conference Series in Mathematics, vol. 103, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, (2005)
Reff, N.: Spectral properties of complex unit gain graphs. Linear Algebra Appl. 436(9), 3165–3176 (2012)
Reff, N., Rusnak, L.J.: An oriented hypergraphic approach to algebraic graph theory. Linear Algebra Appl. 437(9), 2262–2270 (2012)
Rusnak, L.J., Robinson, E., Schmidt, M., Shroff, P.: Oriented hypergraphic matrix-tree type theorems and bidirected minors via boolean ideals, J. Algebr Comb. (2018)
Rusnak, L.J.: Oriented hypergraphs: introduction and balance. Electron. J. Comb. 20(3), 48 (2013)
Rydeheard, D.E., Burstall, R.M.: Computational category theory, Prentice Hall International Series in Computer Science, Prentice Hall International, (1988)
Schiffler, R.: Quiver representations. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, Cham (2014)
Shi, C.J.: A signed hypergraph model of the constrained via minimization problem. Microelectron. J. 23(7), 533–542 (1992)
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Grilliette, W., Rusnak, L.J. Incidence hypergraphs: the categorical inconsistency of set-systems and a characterization of quiver exponentials. J Algebr Comb 58, 1–36 (2023). https://doi.org/10.1007/s10801-023-01232-8
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DOI: https://doi.org/10.1007/s10801-023-01232-8