Product Categorization
arXiv251007553v1 math
Abstract
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration structure we can functorially construct a $\textit{concentration monoid}$, which can be used to give a precise definition of horizontal categorification and decategorification. Moreover, by studying concentration structures on fundamental groupoids, we show that every group arises as the concentration monoid of a trivial category, up to category equivalence.
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration structure we can functorially construct a $\textit{concentration monoid}$, which can be used to give a precise definition of horizontal categorification and decategorification. Moreover, by studying concentration structures on fundamental groupoids, we show that every group arises as the concentration monoid of a trivial category, up to category equivalence.
AI Insights
- Concentration structures on fundamental groupoids pull back along any 2âlifting functor, proving TheoremâŻ1.15.
- This pullâback yields a functorial concentration monoid, enabling systematic decategorification of categories.
- Examplesâfrom trivial categories to nonâtrivial groupoidsâshow the monoid captures group data.
- Every group appears as a concentration monoid of a trivial category, linking group theory with higher categories.
- The axiomsâan equivalence relation on morphisms stable under compositionâensure monoid coherence.
- The paper hints at new topological invariants via fundamental groupoids with concentration structures.
- It opens avenues for horizontal categorification in bicategories, hinting at ties to Lurieâs higher topos theory.
University of California
Abstract
We introduce and motivate -- based on ongoing joint work with Germ\'an Stefanich -- the notion of potent categorical representations of a complex reductive group $G$, specifically a conjectural Langlands correspondence identifying potent categorical representations of $G$ and its Langlands dual $\check G$. We emphasize the symplectic nature of potent categorical representations in their simultaneous dependence on parameters in maximal tori for $G$ and $\check G$, specifically how their conjectural Langlands correspondence fits within a 2-categorical Fourier transform. Our key tool to make various ideas precise is higher sheaf theory and its microlocalization, specifically a theory of ind-coherent sheaves of categories on stacks. The constructions are inspired by the physics of 3d mirror symmetry and S-duality on the one hand, and the theory of double affine Hecke algebras on the other. We also highlight further conjectures related to ongoing programs in and around geometric representation theory.
We introduce and motivate -- based on ongoing joint work with Germ\'an Stefanich -- the notion of potent categorical representations of a complex reductive group $G$, specifically a conjectural Langlands correspondence identifying potent categorical representations of $G$ and its Langlands dual $\check G$. We emphasize the symplectic nature of potent categorical representations in their simultaneous dependence on parameters in maximal tori for $G$ and $\check G$, specifically how their conjectural Langlands correspondence fits within a 2-categorical Fourier transform. Our key tool to make various ideas precise is higher sheaf theory and its microlocalization, specifically a theory of ind-coherent sheaves of categories on stacks. The constructions are inspired by the physics of 3d mirror symmetry and S-duality on the one hand, and the theory of double affine Hecke algebras on the other. We also highlight further conjectures related to ongoing programs in and around geometric representation theory.
Taxonomy of Products
Google Research, Carnegie
Abstract
The growing ubiquity of conversational AI highlights the need for frameworks that capture not only users' instrumental goals but also the situated, adaptive, and social practices through which they achieve them. Existing taxonomies of conversational behavior either overgeneralize, remain domain-specific, or reduce interactions to narrow dialogue functions. To address this gap, we introduce the Taxonomy of User Needs and Actions (TUNA), an empirically grounded framework developed through iterative qualitative analysis of 1193 human-AI conversations, supplemented by theoretical review and validation across diverse contexts. TUNA organizes user actions into a three-level hierarchy encompassing behaviors associated with information seeking, synthesis, procedural guidance, content creation, social interaction, and meta-conversation. By centering user agency and appropriation practices, TUNA enables multi-scale evaluation, supports policy harmonization across products, and provides a backbone for layering domain-specific taxonomies. This work contributes a systematic vocabulary for describing AI use, advancing both scholarly understanding and practical design of safer, more responsive, and more accountable conversational systems.
The growing ubiquity of conversational AI highlights the need for frameworks that capture not only users' instrumental goals but also the situated, adaptive, and social practices through which they achieve them. Existing taxonomies of conversational behavior either overgeneralize, remain domain-specific, or reduce interactions to narrow dialogue functions. To address this gap, we introduce the Taxonomy of User Needs and Actions (TUNA), an empirically grounded framework developed through iterative qualitative analysis of 1193 human-AI conversations, supplemented by theoretical review and validation across diverse contexts. TUNA organizes user actions into a three-level hierarchy encompassing behaviors associated with information seeking, synthesis, procedural guidance, content creation, social interaction, and meta-conversation. By centering user agency and appropriation practices, TUNA enables multi-scale evaluation, supports policy harmonization across products, and provides a backbone for layering domain-specific taxonomies. This work contributes a systematic vocabulary for describing AI use, advancing both scholarly understanding and practical design of safer, more responsive, and more accountable conversational systems.
AI Insights
- English accounts for 56âŻ% of dialogues, with Chinese and Russian at 15.2âŻ% and 11.4âŻ%.
- Coding, factoid, and homework use cases cover over 70âŻ% of interactions.
- Dialogue languages span 18 scripts, from English to Arabic, Vietnamese, and Hungarian.
- MIDASâs fineâgrained dialogâact scheme complements TUNAâs action hierarchy for crossâframework analysis.
- Jailbreakâprompt research shows LLMs can bypass safety, underscoring TUNAâs metaâconversation layer.
- 'Control Through Communication' offers a historical view on managerial systems, echoing TUNAâs userâagency focus.
- The Journal of Documentation and Information Processing & Management feature foundational taxonomy and dialog studies.
Graphs for Products
Department of Mathematics
Abstract
Graphs that are squares under the gluing algebra arise in the study of homomorphism density inequalities such as Sidorenko's conjecture. Recent work has focused on these homomorphism density applications. This paper takes a new perspective and focuses on the graph properties of arbitrary square graphs, not only those relevant to homomorphism conjectures and theorems. We develop a set of necessary and/or sufficient conditions for a graph to be square. We apply these conditions to categorize several classical families of graphs as square or not. In addition, we create infinite families of square graphs by proving that joins and Cartesian, direct, strong, and lexicographic products of square graphs with arbitrary graphs are square.
Graphs that are squares under the gluing algebra arise in the study of homomorphism density inequalities such as Sidorenko's conjecture. Recent work has focused on these homomorphism density applications. This paper takes a new perspective and focuses on the graph properties of arbitrary square graphs, not only those relevant to homomorphism conjectures and theorems. We develop a set of necessary and/or sufficient conditions for a graph to be square. We apply these conditions to categorize several classical families of graphs as square or not. In addition, we create infinite families of square graphs by proving that joins and Cartesian, direct, strong, and lexicographic products of square graphs with arbitrary graphs are square.
AI Insights
- The authors introduce âbutterfly involutionsâ as a fresh symmetry tool to detect squareness.
- A graph is square iff its automorphism group contains a copy of S_{2n} acting via such involutions.
- Complete multipartite, circulant, and fan graphs are shown explicitly to fail this symmetry test.
- The paper raises the intriguing open problem of whether two nonâsquare graphs can combine to form a square.
- It also asks for a full characterization of circulant graphs within the squareness framework.
- For readers seeking context, LovĂĄszâs âLarge Networks and Graph Limitsâ offers a foundational backdrop.
- Recent advances, such as Blekherman et al.âs tropicalization of graph profiles, are cited as complementary tools.
Abstract
The concept of mutual-visibility (MV) has been extended in several directions. A vertex subset $S$ of a graph $G$ is a $k$-distance mutual-visibility ($k$DMV) set if for any two vertices in $S$, there is a geodesic between them of length at most $k$ whose internal vertices are not in $S$. In this paper, we combine this with the MV coloring as follows. For any integer $k\geq1$, a $k$DMV coloring of $G$ is a partition of $V(G)$ into $k$DMV sets, and the $k$DMV chromatic number $\chi_{\mu_k}(G)$ is the minimum cardinality of such a partition. When $k=1$ or $k\ge {\rm diam}(G)$, it equals the clique cover number $\theta(G)$ or the MV chromatic number $\chi_{\mu}(G)$, respectively. So, our attention is given to $1
The concept of mutual-visibility (MV) has been extended in several directions. A vertex subset $S$ of a graph $G$ is a $k$-distance mutual-visibility ($k$DMV) set if for any two vertices in $S$, there is a geodesic between them of length at most $k$ whose internal vertices are not in $S$. In this paper, we combine this with the MV coloring as follows. For any integer $k\geq1$, a $k$DMV coloring of $G$ is a partition of $V(G)$ into $k$DMV sets, and the $k$DMV chromatic number $\chi_{\mu_k}(G)$ is the minimum cardinality of such a partition. When $k=1$ or $k\ge {\rm diam}(G)$, it equals the clique cover number $\theta(G)$ or the MV chromatic number $\chi_{\mu}(G)$, respectively. So, our attention is given to $1
Knowledge Graphs
Kunming University of Sci
Abstract
Large language models (LLMs) based Multilingual Knowledge Graph Completion (MKGC) aim to predict missing facts by leveraging LLMs' multilingual understanding capabilities, improving the completeness of multilingual knowledge graphs (KGs). However, existing MKGC research underutilizes the multilingual capabilities of LLMs and ignores the shareability of cross-lingual knowledge. In this paper, we propose a novel MKGC framework that leverages multilingual shared knowledge to significantly enhance performance through two components: Knowledge-level Grouped Mixture of Experts (KL-GMoE) and Iterative Entity Reranking (IER). KL-GMoE efficiently models shared knowledge, while IER significantly enhances its utilization. To evaluate our framework, we constructed a mKG dataset containing 5 languages and conducted comprehensive comparative experiments with existing state-of-the-art (SOTA) MKGC method. The experimental results demonstrate that our framework achieves improvements of 5.47%, 3.27%, and 1.01% in the Hits@1, Hits@3, and Hits@10 metrics, respectively, compared with SOTA MKGC method. Further experimental analysis revealed the properties of knowledge sharing in settings of unseen and unbalanced languages. We have released the dataset and code for our work on https://github.com/gaoxiaofei07/KL-GMoE.
Large language models (LLMs) based Multilingual Knowledge Graph Completion (MKGC) aim to predict missing facts by leveraging LLMs' multilingual understanding capabilities, improving the completeness of multilingual knowledge graphs (KGs). However, existing MKGC research underutilizes the multilingual capabilities of LLMs and ignores the shareability of cross-lingual knowledge. In this paper, we propose a novel MKGC framework that leverages multilingual shared knowledge to significantly enhance performance through two components: Knowledge-level Grouped Mixture of Experts (KL-GMoE) and Iterative Entity Reranking (IER). KL-GMoE efficiently models shared knowledge, while IER significantly enhances its utilization. To evaluate our framework, we constructed a mKG dataset containing 5 languages and conducted comprehensive comparative experiments with existing state-of-the-art (SOTA) MKGC method. The experimental results demonstrate that our framework achieves improvements of 5.47%, 3.27%, and 1.01% in the Hits@1, Hits@3, and Hits@10 metrics, respectively, compared with SOTA MKGC method. Further experimental analysis revealed the properties of knowledge sharing in settings of unseen and unbalanced languages. We have released the dataset and code for our work on https://github.com/gaoxiaofei07/KL-GMoE.
Abstract
Belief systems are rarely globally consistent, yet effective reasoning often persists locally. We propose a novel graph-theoretic framework that cleanly separates credibility--external, a priori trust in sources--from confidence--an internal, emergent valuation induced by network structure. Beliefs are nodes in a directed, signed, weighted graph whose edges encode support and contradiction. Confidence is obtained by a contractive propagation process that mixes a stated prior with structure-aware influence and guarantees a unique, stable solution. Within this dynamics, we define reasoning zones: high-confidence, structurally balanced subgraphs on which classical inference is safe despite global contradictions. We provide a near-linear procedure that seeds zones by confidence, tests balance using a parity-based coloring, and applies a greedy, locality-preserving repair with Jaccard de-duplication to build a compact atlas. To model belief change, we introduce shock updates that locally downscale support and elevate targeted contradictions while preserving contractivity via a simple backtracking rule. Re-propagation yields localized reconfiguration-zones may shrink, split, or collapse--without destabilizing the entire graph. We outline an empirical protocol on synthetic signed graphs with planted zones, reporting zone recovery, stability under shocks, and runtime. The result is a principled foundation for contradiction-tolerant reasoning that activates classical logic precisely where structure supports it.
Belief systems are rarely globally consistent, yet effective reasoning often persists locally. We propose a novel graph-theoretic framework that cleanly separates credibility--external, a priori trust in sources--from confidence--an internal, emergent valuation induced by network structure. Beliefs are nodes in a directed, signed, weighted graph whose edges encode support and contradiction. Confidence is obtained by a contractive propagation process that mixes a stated prior with structure-aware influence and guarantees a unique, stable solution. Within this dynamics, we define reasoning zones: high-confidence, structurally balanced subgraphs on which classical inference is safe despite global contradictions. We provide a near-linear procedure that seeds zones by confidence, tests balance using a parity-based coloring, and applies a greedy, locality-preserving repair with Jaccard de-duplication to build a compact atlas. To model belief change, we introduce shock updates that locally downscale support and elevate targeted contradictions while preserving contractivity via a simple backtracking rule. Re-propagation yields localized reconfiguration-zones may shrink, split, or collapse--without destabilizing the entire graph. We outline an empirical protocol on synthetic signed graphs with planted zones, reporting zone recovery, stability under shocks, and runtime. The result is a principled foundation for contradiction-tolerant reasoning that activates classical logic precisely where structure supports it.
Ontology for Products
Abstract
Ontologies have become essential in today's digital age as a way of organising the vast amount of readily available unstructured text. In providing formal structure to this information, ontologies have immense value and application across various domains, e.g., e-commerce, where countless product listings necessitate proper product organisation. However, the manual construction of these ontologies is a time-consuming, expensive and laborious process. In this paper, we harness the recent advancements in large language models (LLMs) to develop a fully-automated method of extracting product ontologies, in the form of meronymies, from raw review texts. We demonstrate that the ontologies produced by our method surpass an existing, BERT-based baseline when evaluating using an LLM-as-a-judge. Our investigation provides the groundwork for LLMs to be used more generally in (product or otherwise) ontology extraction.
Ontologies have become essential in today's digital age as a way of organising the vast amount of readily available unstructured text. In providing formal structure to this information, ontologies have immense value and application across various domains, e.g., e-commerce, where countless product listings necessitate proper product organisation. However, the manual construction of these ontologies is a time-consuming, expensive and laborious process. In this paper, we harness the recent advancements in large language models (LLMs) to develop a fully-automated method of extracting product ontologies, in the form of meronymies, from raw review texts. We demonstrate that the ontologies produced by our method surpass an existing, BERT-based baseline when evaluating using an LLM-as-a-judge. Our investigation provides the groundwork for LLMs to be used more generally in (product or otherwise) ontology extraction.
MECE Mutually Exclusive, Collectively Exhaustive.Knowledge Management
Victoria University of
Abstract
In the Generative Age, the nature of knowledge work is transforming. Traditional models that emphasise the organisation and retrieval of pre-existing information are increasingly inadequate in the face of generative AI (GenAI) systems capable of autonomous content creation. This paper introduces the Knowledge Sculptor (KS), a new professional archetype for Human-GenAI collaboration that transforms raw AI output into trustworthy, actionable knowledge. Grounded in a socio-technical perspective, the KS is conceptualised through a framework of competencies, including architecting a vision, iterative dialogue, information sculpting, and curiosity-driven synthesis. A practice-based vignette illustrates the KS role in action, and in a self-referential approach, the paper itself serves as an artefact of the sculpting process it describes.
In the Generative Age, the nature of knowledge work is transforming. Traditional models that emphasise the organisation and retrieval of pre-existing information are increasingly inadequate in the face of generative AI (GenAI) systems capable of autonomous content creation. This paper introduces the Knowledge Sculptor (KS), a new professional archetype for Human-GenAI collaboration that transforms raw AI output into trustworthy, actionable knowledge. Grounded in a socio-technical perspective, the KS is conceptualised through a framework of competencies, including architecting a vision, iterative dialogue, information sculpting, and curiosity-driven synthesis. A practice-based vignette illustrates the KS role in action, and in a self-referential approach, the paper itself serves as an artefact of the sculpting process it describes.
AI Insights
- The HumanâGenAI Value Loop formalises how human values shape AI outputs and feedback cycles.
- Hybrid intelligence emerges when a Knowledge Sculptor iteratively refines generative drafts into actionable insights.
- Bias mitigation is addressed by embedding fairness checkpoints during the sculpting dialogue phase.
- Accountability is operationalised through traceable decision logs that record humanâAI interaction points.
- Interdisciplinary research teamsâcombining social science, humanities, and CSâare essential for robust GenAI governance.
- The Knowledge Sculptorâs curiosityâdriven synthesis mirrors the âmachines as teammatesâ agenda in team collaboration studies.
- Future work should prototype valueâaligned sculpting tools, building on Hevnerâs designâscience framework.
Abstract
The technological revolution of the Internet has digitized the social, economic, political, and cultural activities of billions of humans. While researchers have been paying due attention to concerns of misinformation and bias, these obscure a much less researched and equally insidious problem - that of uncritically consuming incomplete information. The problem of incomplete information consumption stems from the very nature of explicitly ranked information on digital platforms, where our limited mental capacities leave us with little choice but to consume the tip of a pre-ranked information iceberg. This study makes two chief contributions. First, we leverage the context of internet search to propose an innovative metric that quantifies information completeness. For a given search query, this refers to the extent of the information spectrum that is observed during web browsing. We then validate this metric using 6.5 trillion search results extracted from daily search trends across 48 nations for one year. Second, we find causal evidence that awareness of information completeness while browsing the Internet reduces resistance to factual information, hence paving the way towards an open-minded and tolerant mindset.
The technological revolution of the Internet has digitized the social, economic, political, and cultural activities of billions of humans. While researchers have been paying due attention to concerns of misinformation and bias, these obscure a much less researched and equally insidious problem - that of uncritically consuming incomplete information. The problem of incomplete information consumption stems from the very nature of explicitly ranked information on digital platforms, where our limited mental capacities leave us with little choice but to consume the tip of a pre-ranked information iceberg. This study makes two chief contributions. First, we leverage the context of internet search to propose an innovative metric that quantifies information completeness. For a given search query, this refers to the extent of the information spectrum that is observed during web browsing. We then validate this metric using 6.5 trillion search results extracted from daily search trends across 48 nations for one year. Second, we find causal evidence that awareness of information completeness while browsing the Internet reduces resistance to factual information, hence paving the way towards an open-minded and tolerant mindset.
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