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Hypergraph Categories [article]

Brendan Fong, David I Spivak
2019 arXiv   pre-print
First is a coherence theorem for hypergraph categories, which says that every hypergraph category is equivalent to an objectwise-free hypergraph category.  ...  Second, we prove that the category of objectwise-free hypergraph categories is equivalent to the category of cospan-algebras.  ...  Strictification of hypergraph categories In this subsection we prove that every hypergraph category is hypergraph equivalent to a strict hypergraph category.  ... 
arXiv:1806.08304v3 fatcat:vqwwgtw2knacbpcqku45rpyf3i

Hypergraph Basic Categorial Grammars [chapter]

Tikhon Pshenitsyn
2020 Lecture Notes in Computer Science  
We consider a particular approach called basic categorial grammar (BCG) and introduce its natural extension to hypergraphs -hypergraph basic categorial grammar (HBCG).  ...  This work is an attempt to generalize categorial grammars, which deal with string languages, to hypergraphs.  ...  Rule (÷) The concept of hypergraph basic categorial grammars (HBCGs) is based on the mechanism of reduction of hypergraphs labeled by types.  ... 
doi:10.1007/978-3-030-51372-6_9 fatcat:wmlgbavygvbzdi52oigphwb6sy

Rewriting in Free Hypergraph Categories

Fabio Zanasi
2017 Electronic Proceedings in Theoretical Computer Science  
This interpretation allows to use results on hypergraphs to ensure decidability of confluence for rewriting in a free hypergraph category.  ...  These categories, also called hypergraph categories, are increasingly relevant: Frobenius structures recently appeared in cross-disciplinary applications, including the study of quantum processes, dynamical  ...  We write ⊕ for the monoidal product in a monoidal category. Props and Hypergraph Categories We are going to study hypergraph categories freely generated by a signature of operations.  ... 
doi:10.4204/eptcs.263.2 fatcat:meiolnyhvfhafci7ymy65jjmle

Finite matrices are complete for (dagger-)hypergraph categories [article]

Aleks Kissinger
2015 arXiv   pre-print
Dagger-hypergraph categories are the same, but with dagger-symmetric monoidal categories and dagger-SCFAs.  ...  Hypergraph categories are symmetric monoidal categories where each object is equipped with a special commutative Frobenius algebra (SCFA).  ...  This becomes a dagger-hypergraph category in the obvious way, and by construction, forms the free dagger-hypergraph category.  ... 
arXiv:1406.5942v2 fatcat:fm2meueyobftjnzqsbu2xgowbm

Weak ω-categories as ω-hypergraphs [article]

Hiroyuki Miyoshi, Toru Tsujishita
2000 arXiv   pre-print
In this paper, firstly, we introduce a higher-dimensional analogue of hypergraphs, namely ω-hypergraphs.  ...  The whole process of our definition is in parallel with the way of regarding categories as graphs with composition and identities.  ...  This notion contains various categorical algebras: ω-categories, bicategoreis, double categories, etc. Meanwhile, authors noticed that it can be thought of as a form of ω-dimensional hypergraphs.  ... 
arXiv:math/0003137v1 fatcat:pidtmcsy6jfrvadnacbjq2rkae

Exponential Objects in Categories of Generalized Uniform Hypergraphs [article]

Martin Schmidt
2018 arXiv   pre-print
We construct exponential objects in categories of generalized uniform hypergraphs and use embeddings induced by nerve-realization adjunctions to show why conventional categories of graphs and hypergraphs  ...  Interpretations in Conventional Categories More conventional categories of graphs, uniform hypergraphs, and hypergraphs can be given as certain comma categories.  ...  Introduction It is well-known that exponential objects do not exist in conventional categories of uniform hypergraphs.  ... 
arXiv:1807.09345v1 fatcat:j5gbf4glkbgwtcm5vbcdvaqs3a

Injectives Hulls and Projective Covers in Categories of Generalized Uniform Hypergraphs [article]

Martin Schmidt
2018 arXiv   pre-print
We construct injective hulls and projective covers in categories of generalized uniform hypergraphs which generalizes the constructions in the category of quivers and the category of undirected graphs.  ...  We are able to use the adjunction above to classify the projective objects in the category of hypergraphs.  ...  In other words, we allow infinite vertex and edge sets, multiple edges, loops, empty edges and empty vertices. 7 In other words the category of hypergraphs H is the category of P-graphs.  ... 
arXiv:1807.09348v1 fatcat:i677kpydyndhfn34uiyqnwi6ci

Custom Hypergraph Categories via Generalized Relations [article]

Dan Marsden, Fabrizio Genovese
2017 pre-print
The resulting categories are preorder-enriched and provide analogues of relational converse and taking graphs of maps.  ...  The category Rel (Σ,E) (Q) is a hypergraph category.  ...  Definition 3 (Hypergraph Category).  ... 
doi:10.4230/lipics.calco.2017.17 arXiv:1703.01204v1 fatcat:esthpsnw7fhnpmlzgag7hogekm

Construction of Pushout Complements in the Category of Hypergraphs

Marvin Heumüller, Salil Joshi, Barbara König, Jan Stückrath
2011
We describe a concrete construction of all pushout complements for two given morphisms f : A -> B, m: B -> D in the category of hypergraphs, valid also for the case where f, m are non-injective.  ...  Conclusion We have shown how to construct pushout complements in the category of hypergraphs in the general case when both given morphisms might be non-injective.  ...  Definition 2 (Hypergraph morphism). Let G, G be (Λ-)hypergraphs.  ... 
doi:10.14279/tuj.eceasst.39.647.667 fatcat:hksy63m6qrgeliwq7fm64ae22m

Take-Away Impartial Combinatorial Games on Hypergraphs and Other Related Geometric and Discrete Structures [article]

T. H. Molena
2022 arXiv   pre-print
Games on neither Oddly nor Evenly Uniform Hypergraphs.  ...  In a Take-Away Game on hypergraphs, two players take turns to remove the vertices and the hyperedges of the hypergraphs. In each turn, a player must remove either a single vertex or a hyperedge.  ...  hypergraph with only hyperedges (with all vertices in those hyperedges) in category Y, and 0 hyperedge in category X is an oddly uniform hypergraph.  ... 
arXiv:2203.09696v1 fatcat:wnkjsxk2yjdo3j5d6hag3rkmlq

Infinite hypergraphs I. Basic properties

Michael Bauderon
1991 Theoretical Computer Science  
Basic properties of the category of infinite directed hyperedge-iabeiied hypergraphs are studied.  ...  Rkume' On Ctudie ies propriCtb fondamentaies des hypergraphes infinis & hyperar&s Ctiquetees.  ...  V = V*) where E and V are two sets, i.e. the category of directed unlabelled hypergraphs (the arrows of the comma category are actually hypergraph morphisms).  ... 
doi:10.1016/0304-3975(91)90222-n fatcat:pgk4emb4ufcjdklmioobuyjf7i

A recipe for black box functors [article]

Brendan Fong, Maru Sarazola
2018 arXiv   pre-print
One way to solve these problems is to formalise the diagrams and their semantics using hypergraph categories, with semantic interpretation a hypergraph functor, called the black box functor, between them  ...  Reviewing a principled method for constructing hypergraph categories and functors, known as decorated corelations, in this paper we construct a category of decorating data, and show that the decorated  ...  From hypergraph categories to cospan algebras Given a hypergraph category H, we can construct a cospan algebra in two steps.  ... 
arXiv:1812.03601v1 fatcat:w2pivxeekffmdhxcmlabvibhzq

Scene Classification via Hypergraph-Based Semantic Attributes Subnetworks Identification [chapter]

Sun-Wook Choi, Chong Ho Lee, In Kyu Park
2014 Lecture Notes in Computer Science  
By searching subnetworks on a hypergraph, we extract the interaction subnetworks of the semantic attributes that are optimized for classifying individual scene categories.  ...  In this paper, we propose a novel hypergraph-based modeling that considers the higher-order relationship of semantic attributes in a scene and apply it to scene classification.  ...  Then, for each scene category, category-specific semantic attribute (CSSA) subnetworks are obtained by proposed hypergraph based method.  ... 
doi:10.1007/978-3-319-10584-0_24 fatcat:zowmavuxl5ar5jgosorb4banta

The representation of semantic similarities between object concepts in the brain: a hypergraph-based model

Kaoutar Skiker, Mounir Maouene
2014 BMC Neuroscience  
Third, in a hypergraph, the clusters of objects overlap, because one category can share one or more of its semantic features with one or more categories and, by extension, a group of categories.  ...  The benefits of using hypergraphs for representing object similarities are threefold.  ...  Third, in a hypergraph, the clusters of objects overlap, because one category can share one or more of its semantic features with one or more categories and, by extension, a group of categories.  ... 
doi:10.1186/1471-2202-15-s1-p84 pmcid:PMC4126571 fatcat:uanpagtmcfebxbw6ll54kicjsi

Vertex-Weighted Hypergraph Learning for Multi-View Object Classification

Lifan Su, Yue Gao, Xibin Zhao, Hai Wan, Ming Gu, Jiaguang Sun
2017 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence  
To this end, we propose a vertex-weighted hypergraph learning algorithm for multi-view 3D object classification, introducing an updated hypergraph structure.  ...  In this work, we propose to employ the hypergraph structure to formulate the relationship among 3D objects, taking the advantage of hypergraph on high-order correlation modelling.  ...  The relevance matrix F for hypergraph G = (V, E, W) is the to-be-learned matrix, which indicates the relevance of each object to the categories.  ... 
doi:10.24963/ijcai.2017/387 dblp:conf/ijcai/SuGZW0S17 fatcat:jt7e23sehng4hhwytnrk26ndfa
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