Language Modeling with Reduced Densities

This work originates from the observation that today's state-of-the-art statistical language models are impressive not only for their performance, but also---and quite crucially---because they are built entirely from correlations in unstructured text data. The latter observation prompts a fundamental question that lies at the heart of this paper: What mathematical structure exists in unstructured text data? We put forth enriched category theory as a natural answer. We show that sequences of symbols from a finite alphabet, such as those found in a corpus of text, form a category enriched over probabilities. We then address a second fundamental question: How can this information be stored and modeled in a way that preserves the categorical structure? We answer this by constructing a functor from our enriched category of text to a particular enriched category of reduced density operators. The latter leverages the Loewner order on positive semidefinite operators, which can further be interpreted as a toy example of entailment.

Metric monads

We develop universal algebra over an enriched category and relate it to finitary enriched monads over . Using it, we deduce recent results about ordered universal algebra where inequations are used instead of equations. Then we apply it to metric universal algebra where quantitative equations are used instead of equations. This contributes to understanding of finitary monads on the category of metric spaces.

Enriched category as a model of qualia structure based on similarity judgements

Qualitative relationships between two instances of conscious experiences can be quantified through the perceived similarity. Previously, we proposed that by defining similarity relationships as arrows and conscious experiences as objects, we can define a category of qualia in the context of category theory. However, the example qualia categories we proposed were highly idealized and limited to cases where perceived similarity is binary: either present or absent without any gradation. Here, we introduce enriched category theory to address the graded levels of similarity that arises in many instances of qualia. Enriched categories generalize the concept of a relation between objects as a directed arrow (or morphism) in ordinary category theory to a more flexible notion, such as a measure of distance. As an alternative relation, here we propose a graded measure of perceived dissimilarity between the two objects. We claim that enriched categories accommodate various types of conscious experiences. An important extension of this claim is the application of the Yoneda lemma in enriched category; we can characterize a quale through a collection of relationships between the quale and the other qualia up to an (enriched) isomorphism.

Cartesian Differential Categories as Skew Enriched Categories

We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

Enriched category as a model of qualia structure based on similarity judgements

Qualitative relationships between two instances of conscious experiences can be quantified through the perceived similarity. Previously, we proposed that by defining similarity relationships as arrows and conscious experiences as objects, we can define a category of qualia in the context of category theory. However, the example qualia categories we proposed were highly idealized and limited to cases where perceived similarity is binary: either present or absent without any gradation. When similarity is graded, a situation can arise where A0 is similar to A1, A1 is similar to A2, and so on, yet A0 is not similar to An, which is called the Sorites paradox. Here, we introduce enriched category theory to address this situation. Enriched categories generalize the concept of a relation between objects as a directed arrow (or morphism) in ordinary category theory to a more flexible notion, such as a measure of distance. As an alternative relation, here we propose a graded measure of perceived dissimilarity between the two objects. These measures combine in a way that addresses the Sorites paradox; even if the dissimilarity between Ai and Ai+1 is small for i = 0 … n, hence perceived as similar, the dissimilarity between A0 and An can be large, hence perceived as different. In this way, we show how dissimilarity-enriched categories of qualia resolve the Sorites paradox. We claim that enriched categories accommodate various types of conscious experiences. An important extension of this claim is the application of the Yoneda lemma in enriched category; we can characterize a quale through a collection of relationships between the quale and the other qualia up to an (enriched) isomorphism.

Insight into the mechanisms of terminal HS-tolerance in wheat mutant with improved nutritional quality through de novo transcriptome sequencing

Terminal HS is one of the main bottle-neck in wheat yield and grain-quality. Here, we have developed wheat mutant (M3) for HS-tolerance [parent-MP3054- C-306/CB.SPRING BW/CPAN2072 (Parentage)]. To elucidate the mechanism of thermotolerance in mutant, we performed de novo transcriptomic sequencing of mutant (M3), parent (P3), and mutant exposed to HS (M3H). We sequenced 6.5, 7.5, and 7.0 million reads in P3, M3 and M3H and generated 3,05,537 genes and 5,88,788 transcripts with an N50 of 1,349 bp. We observed 6,120 upregulated and 4,428 downregulated transcripts (M3 vs P3), 11,354 upregulated and 12,408 downregulated genes (M3H vs P3) and 4817 upregulated and 9085 downregulated genes (M3H vs M3). Some of the highly upregulated genes observed were HSP20, SOD, ABC transporters, HSF, etc. and downregulated genes were starch synthase, sucrose synthase, etc. Gene Ontology analysis showed ‘ATP-binding’ to be most enriched category. Carbon metabolism pathway was observed most altered under HS. We identified 41940 SSRs, 1,10,772 SNPs and 2432 InDels. Potential markers were observed lying on HSP, SOD, STK, and starch synthase. Biochemical markers based characterization showed wheat mutant to be better in HS-tolerance and grain-quality, as compared to parent.

THH and Traces of Enriched Categories

We prove that topological Hochschild homology (THH) arises from a presheaf of circles on a certain combinatorial category, which gives a universal construction of THH for any enriched $\infty $-category. Our results rely crucially on an elementary, model-independent framework for enriched higher-category theory, which may be of independent interest. For those interested only in enriched category theory, read Sections 1.3 and 2.

What does it mean to enrich the definition of consciousness in consciousness research using (enriched) category theory?

In our recent essay on Cognitive Science [Naotsugu Tsuchiya & Hayato Saigo (2019) “Understanding Consciousness Through Category Theory” vol 26, pp 462 - 477], we provided a general introduction of category theory to consciousness researchers. Further, we also provided our tentative theoretical sketches on our latest ideas on how to apply tools in category theory into consciousness research. In particular, we discussed how we can propose categories of level of consciousness and categories of contents of consciousness. We also speculated what (if any) these efforts will bring into consciousness research. In this short piece, we will address several comments we received on our essay on the same volume from six experts, providing some clarification on three issues: 1) significance of our proposal of a novel viewpoint to enrich what it means to define consciousness, 2) possibility of category theoretical interpretation of consciousness, and 3) understanding of consciousness through the enriched category theoretical framework.