scholarly journals Approximate injectivity and smallness in metric-enriched categories

2021 ◽  
pp. 106974
Author(s):  
J. Adámek ◽  
J. Rosický
Keyword(s):  
Enriched Categories

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.

scholarly journals Genome-wide transcriptional changes triggered by water deficit on a drought-tolerant common bean cultivar

2020 ◽  
Vol 20 (1) ◽  
Author(s):  
Josefat Gregorio Jorge ◽  
Miguel Angel Villalobos-López ◽  
Karen Lizeth Chavarría-Alvarado ◽  
Selma Ríos-Meléndez ◽  
Melina López-Meyer ◽  
...  
Keyword(s):  
Cell Wall ◽  
Drought Tolerance ◽  
Common Bean ◽  
Crop Production ◽  
Molecular Mechanisms ◽  
Molecular Data ◽  
Transcriptional Level ◽  
Cell Periphery ◽  
Enriched Categories ◽  
Drought Tolerant

Abstract Background Common bean (Phaseolus vulgaris L.) is a relevant crop cultivated over the world, largely in water insufficiency vulnerable areas. Since drought is the main environmental factor restraining worldwide crop production, efforts have been invested to amend drought tolerance in commercial common bean varieties. However, scarce molecular data are available for those cultivars of P. vulgaris with drought tolerance attributes. Results As a first approach, Pinto Saltillo (PS), Azufrado Higuera (AH), and Negro Jamapa Plus (NP) were assessed phenotypically and physiologically to determine the outcome in response to drought on these common bean cultivars. Based on this, a Next-generation sequencing approach was applied to PS, which was the most drought-tolerant cultivar to determine the molecular changes at the transcriptional level. The RNA-Seq analysis revealed that numerous PS genes are dynamically modulated by drought. In brief, 1005 differentially expressed genes (DEGs) were identified, from which 645 genes were up-regulated by drought stress, whereas 360 genes were down-regulated. Further analysis showed that the enriched categories of the up-regulated genes in response to drought fit to processes related to carbohydrate metabolism (polysaccharide metabolic processes), particularly genes encoding proteins located within the cell periphery (cell wall dynamics). In the case of down-regulated genes, heat shock-responsive genes, mainly associated with protein folding, chloroplast, and oxidation-reduction processes were identified. Conclusions Our findings suggest that secondary cell wall (SCW) properties contribute to P. vulgaris L. drought tolerance through alleviation or mitigation of drought-induced osmotic disturbances, making cultivars more adaptable to such stress. Altogether, the knowledge derived from this study is significant for a forthcoming understanding of the molecular mechanisms involved in drought tolerance on common bean, especially for drought-tolerant cultivars such as PS.

Mealy Morphisms of Enriched Categories

2010 ◽  
Vol 20 (3) ◽  
pp. 251-273 ◽  
Author(s):  
Robert Paré
Keyword(s):  
Enriched Categories

Generalized Enrichments of Categories for Operads

2007 ◽  
Vol 14 (01) ◽  
pp. 61-78
Author(s):  
Zizhu Ma
Keyword(s):  
Monoidal Category ◽  
Special Kind ◽  
Category Structure ◽  
Commutative Monoids ◽  
Enriched Categories

Most enriched categories also have an ordinary category structure which is compatible with the enrichment on them. In this paper, enrichments in a monoidal category are generalized to arbitrary categories. These specialize to the classical enrichments when sets are regraded as discrete categories. We also generalize the definitions of PROs and PROPs as some generalized enrichments of categories. Then an operad in some monoidal category corresponds to a generalized PROP. Algebras of operads induce some special kind of monoidal functors. In the category of small categories, we construct several operads to define lax monoids and lax commutative monoids which are formal descriptions of natural associativity and commutativity. Using this identification, operads and their algebras can be studied by lax commutative monoids and morphisms between them.

scholarly journals A duality of quantale-enriched categories

2012 ◽  
Vol 216 (8-9) ◽  
pp. 1866-1878 ◽  
Author(s):  
Dirk Hofmann ◽  
Paweł Waszkiewicz
Keyword(s):  
Enriched Categories

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.

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