[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84328-en":3,"doc-seo-84328-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84328,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Subword Representations and Weak Hypercube Dimension for Acyclic Categories","The paper develops a categorical analogue of weak hypercube representations for finite posets by using faithful embeddings into categories of subwords of finite words. For finite acyclic categories, it characterizes exactly which ones admit such weak subword representations: precisely the monic categories whose hom-sets admit a left-compatible local total order. The construction is explicit, producing a concrete word representation, and it introduces a generalized query game whose winning sets yield upper bounds for weak word dimension.","arXiv :2607 .08210v1 [math .CO] 9 Jul 2026  \nSUBWORD REPRESENTATIONS AND WEAK HYPERCUBE DIMENSION FOR ACYCLIC CATEGORIES  \nISAAC CARCACÍA-CAMPOS  \nAbstract. We introduce a categorical analogue of weak hypercube representations of finite posets by means of faithful embeddings into categories of subwords of finite words. For finite acyclic categories, we characterize those admitting such a weak subword representation: they are precisely the monic categories whose homsets carry a left-compatible local total order. The proof is constructive and gives an explicit word representation. We also introduce a query game for categories, generalizing a Boolean query game for posets, and show how winning sets produce explicit word representations and hence upper bounds for the weak word dimension.  \n1. Introduction  \nThe abstract notion of a partial order encompasses many different kinds of relations. In mereology, for instance, it is usually assumed that the relation being a part of is a partial order, possibly satisfying additional axioms [2] . There are, however, some caveats to this point of view.  \nFor example, consider the word attack. The one-letter words a and t are subwords of attack, but each occurs in two distinct ways: the letter a can be embedded in positions 1 or 4, and t in positions 2 or 3. By contrast, c and k occur only once. Thus, a mere yes/no relation of parthood loses information about the number of possible embeddings. Similar phenomena arise when modelling processes with repeated homogeneous parts, such as music [3] .  \nThis suggests replacing partially ordered sets by richer structures capable of recording not only whether one object is part of another, but also how many different ways this can happen. In this paper we use small acyclic categories for this purpose.  \n2020 Mathematics Subject Classification. Primary: 18B35; Secondary: 06A07, 68R15 .  \nKey words and phrases. acyclic category, subword, hypercube dimension, posets, query game.  \n2 ISAAC CARCACÍA-CAMPOS  \nSuch a category may be viewed as a directed acyclic multigraph equipped with a transitive structure given by composition of morphismsin which we impose some relations [1] . Every poset is a special case: it is a skeletal acyclic category with at most one morphism between any two objects, where reflexivity and transitivity are transmuted into identity morphisms and composition.  \nThe central example for us is the category Subw of subwords of a fixed word w over an alphabet Σ . Its objects are the words that occur as subwords of w , and its morphisms are the possible subword inclusions, i.e. injective order-preserving maps between positions that respect the labels.  \nA particularly important case occurs when all letters of w are distinct. Then Subw is isomorphic to the Boolean lattice 2 [n], where n is the length of w. Thus categories of subwords generalize Boolean lattices by allowing repeated letters, and repeated letters produce parallel morphisms.  \nIn order theory, several notions of dimension have been studied for finite posets. The classical one is the Dushnik–Miller dimension, or order dimension, introduced in [4] . It is the minimum number of linear extensions whose intersection is the given poset. Equivalently, it is the minimum number of chains whose product contains the poset as an order-embedded subposet [11, chapter 10] .  \nAnother invariant is the 2-dimension, or hypercube dimension [5 , 14] . In its standard form, it is the least integer n for which the poset embedsas a subposet of the Boolean lattice 2 [n] . Equivalently, it is the least n for which there exists an order-preserving and order-reflecting map into 2 [n] .  \nIn this paper we work with a weaker version. The weak 2-dimension of a finite poset P , denoted dimw2(P), is the least integer n for which there exists an injective order-preserving map  \nP −→ 2 [n] .  \nUnlike the usual 2-dimension, this weak version does not require incomparability to be reflected. For example, an antich","cbCaikvqHq5hwAc0","https://ap.wps.com/l/cbCaikvqHq5hwAc0","pdf",454901,1,30,"English","en",105,"# Introduction\n## Subword categories and dimension notions\n## Weak word dimension for acyclic categories\n# Main results and constructions\n## Characterization via monic categories and ordered hom-sets\n## Stability under categorical constructions\n## Query game and upper bounds","[{\"question\":\"What is the main goal of the paper?\",\"answer\":\"To define and characterize a categorical version of weak hypercube representations for finite acyclic categories using faithful embeddings into categories of subwords.\"},{\"question\":\"How does the paper define weak word dimension for a small acyclic category?\",\"answer\":\"It is the smallest non-negative integer n such that there exist an alphabet, a word of length n, and a faithful functor from the category into the corresponding subword category, injective on objects.\"},{\"question\":\"What is the paper’s characterization of finite acyclic categories that admit weak subword representations?\",\"answer\":\"They are exactly the finite acyclic categories that are monic and whose hom-sets carry a left-compatible local total order.\"}]",1784194854,76,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"subword-representations-and-weak-hypercube-dimension-for-acyclic-categories","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/subword-representations-and-weak-hypercube-dimension-for-acyclic-categories/84328/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main goal of the paper?","Question",{"text":75,"@type":76},"To define and characterize a categorical version of weak hypercube representations for finite acyclic categories using faithful embeddings into categories of subwords.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper define weak word dimension for a small acyclic category?",{"text":80,"@type":76},"It is the smallest non-negative integer n such that there exist an alphabet, a word of length n, and a faithful functor from the category into the corresponding subword category, injective on objects.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the paper’s characterization of finite acyclic categories that admit weak subword representations?",{"text":84,"@type":76},"They are exactly the finite acyclic categories that are monic and whose hom-sets carry a left-compatible local total 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