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

2021 ◽  
Author(s):  
Naotsugu Tsuchiya ◽  
Steven Phillips ◽  
Hayato Saigo
Keyword(s):  
Category Theory ◽  
Sorites Paradox ◽  
Perceived Similarity ◽  
Enriched Category ◽  
Enriched Categories ◽  
Important Extension ◽  
Similarity Relationships ◽  
Yoneda Lemma ◽  
Similarity Judgements ◽  
Qualia Structure

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.

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

2021 ◽  
Author(s):  
Naotsugu Tsuchiya ◽  
Hayato Saigo ◽  
Steven Phillips
Keyword(s):  
Category Theory ◽  
The Other ◽  
Perceived Similarity ◽  
Enriched Category ◽  
Enriched Categories ◽  
Important Extension ◽  
Similarity Relationships ◽  
Yoneda Lemma ◽  
Similarity Judgements ◽  
Qualia Structure

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.

scholarly journals THH and Traces of Enriched Categories

2020 ◽  
Author(s):  
John D Berman
Keyword(s):  
Category Theory ◽  
Hochschild Homology ◽  
Independent Interest ◽  
Elementary Model ◽  
Enriched Category ◽  
Enriched Categories ◽  
Higher Category Theory ◽  
Model Independent ◽  
Universal Construction ◽  
Topological Hochschild Homology

Abstract 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.

Completions of Non-Symmetric Metric Spaces Via Enriched Categories

2009 ◽  
Vol 16 (1) ◽  
pp. 157-182
Author(s):  
Vincent Schmitt
Keyword(s):  
Category Theory ◽  
Metric Spaces ◽  
Enriched Category ◽  
Closed Category ◽  
Enriched Categories ◽  
Terminal Object ◽  
Base Category ◽  
Monoidal Closed Category ◽  
The One ◽  
Universal Completion

Abstract It is known from [Lawvere, Repr. Theory Appl. Categ. 1: 1–37 2002] that nonsymmetric metric spaces correspond to enrichments over the monoidal closed category [0, ∞]. We use enriched category theory and in particular a generic notion of flatness to describe various completions for these spaces. We characterise the weights of colimits commuting in the base category [0, ∞] with the conical terminal object and cotensors. Those can be interpreted in metric terms as very general filters, which we call filters of type 1. This correspondence extends the one between minimal Cauchy filters and weights which are adjoint as modules. Translating elements of enriched category theory into the metric context, one obtains a notion of convergence for filters of type 1 with a related completeness notion for spaces, for which there exists a universal completion. Another smaller class of flat presheaves is also considered both in the context of both metric spaces and preorders. (The latter being enrichments over the monoidal closed category 2.) The corresponding completion for preorders is the so-called dcpo completion.

scholarly journals A relational approach to consciousness: categories of level and contents of consciousness

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Naotsugu Tsuchiya ◽  
Hayato Saigo
Keyword(s):  
Empirical Research ◽  
Category Theory ◽  
Gold Standard ◽  
Empirical Testing ◽  
Relational Approach ◽  
Alternative Approach ◽  
Key Concepts ◽  
Categorical Structure ◽  
Yoneda Lemma ◽  
Qualia Structure

Abstract Characterizing consciousness in and of itself is notoriously difficult. Here, we propose an alternative approach to characterize, and eventually define, consciousness through exhaustive descriptions of consciousness’ relationships to all other consciousness. This approach is founded in category theory. Indeed, category theory can prove that two objects A and B in a category can be equivalent if and only if all the relationships that A holds with others in the category are the same as those of B; this proof is called the Yoneda lemma. To introduce the Yoneda lemma, we gradually introduce key concepts of category theory to consciousness researchers. Along the way, we propose several possible definitions of categories of consciousness, both in terms of level and contents, through the usage of simple examples. We propose to use the categorical structure of consciousness as a gold standard to formalize empirical research (e.g. color qualia structure at fovea and periphery) and, especially, the empirical testing of theories of consciousness.

scholarly journals Cartesian Differential Categories as Skew Enriched Categories

2021 ◽  
Author(s):  
Richard Garner ◽  
Jean-Simon Pacaud Lemay
Keyword(s):  
Linear Logic ◽  
Usual Sense ◽  
Enriched Category ◽  
Enriched Categories ◽  
Full Structure ◽  
Monoidal Closed Category ◽  
Category Of Modules ◽  
Differential Categories ◽  
Linear Category ◽  
Open Question

AbstractWe 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.

Yoneda lemma for enriched ∞-categories

2020 ◽  
Vol 367 ◽  
pp. 107129
Author(s):  
Vladimir Hinich
Keyword(s):  
Enriched Categories ◽  
Yoneda Lemma

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

2020 ◽  
Author(s):  
Naotsugu Tsuchiya ◽  
Hayato Saigo
Keyword(s):  
Cognitive Science ◽  
Category Theory ◽  
Theoretical Framework ◽  
Theoretical Interpretation ◽  
General Introduction ◽  
Enriched Category ◽  
Level Of Consciousness ◽  
Short Piece ◽  
Definition Of

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.

2-Dimensional Categories

2021 ◽  
Author(s):  
Niles Johnson ◽  
Donald Yau
Keyword(s):  
Tensor Product ◽  
Category Theory ◽  
Basic Category ◽  
Double Categories ◽  
Grothendieck Construction ◽  
Monoidal Bicategories ◽  
Coherence Theorem ◽  
First Time ◽  
Yoneda Lemma

2-Dimensional Categories provides an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories; pasting diagrams; lax functors; 2-/bilimits; the Duskin nerve; the 2-nerve; internal adjunctions; monads in bicategories; 2-monads; biequivalences; the Bicategorical Yoneda Lemma; and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.

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