Kan extensions in Enriched Category Theory

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.

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THH and Traces of Enriched Categories

International Mathematics Research Notices ◽

10.1093/imrn/rnaa238 ◽

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.

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

10.31219/osf.io/87e96 ◽

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.

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