Saunders Mac Lane famously remarked that "Bourbaki just missed" formulating adjoints in a 1948 appendix (written no doubt by Pierre Samuel) to an early draft of Algebre--which then had to wait until Daniel Kan's 1958 paper on adjoint functors. But Mac Lane was using the orthodox treatment of adjoints that only contemplates the object-to-object morphisms within a category, i.e., homomorphisms. When Samuel's treatment is reconsidered in view of the treatment of adjoints using heteromorphisms or hets (object-to-object morphisms between objects (...) in different categories), then he, in effect, isolated the concept of a left representation solving a universal mapping problem. When dualized to obtain the concept of a right representation, the two halves only need to be united to obtain an adjunction. Thus Samuel was only a now-simple dualization away for formulating adjoints in 1948. Apparently, Bodo Pareigis' 1970 text was the first and perhaps only text to give the heterodox "new characterization" (i.e., heteromorphic treatment) of adjoints. Orthodox category theory uses various relatively artificial devices to avoid formally recognizing hets--even though hets are routinely used by the working mathematician. Finally we consider a "philosophical" question as to whether the most important concept in category theory is the notion of an adjunction or the notion of a representation giving a universal mapping property (where adjunctions arise as the special case of a bi-representation of dual universal mapping problems). (shrink)

Graphical AbstractSquamates may be an attractive group in which to study the influence of sex chromosomes on speciation rates because of the repeated evolution of heterogamety (both XY and ZW), as well as an apparently large number of taxa with environmental sex-determination.

There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms that parses an adjunction into two separate parts. Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory (...) and is focused on the interpretation and application of the mathematical concepts. (shrink)

Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunctions being the primary lens. If adjunctions are so important in mathematics, then perhaps they will isolate concepts of some importance in the empirical sciences. But the applications of adjunctions have been hampered by an overly restrictive formulation that avoids heteromorphisms or hets. By reformulating an adjunction using hets, it is split into two parts, a left and a right semiadjunction. (...) Semiadjunctions (essentially a formulation of universal mapping properties using hets) can then be combined in a new way to define the notion of a brain functor that provides an abstract model of the intentionality of perception and action (as opposed to the passive reception of sense-data or the reflex generation of behavior). (shrink)

Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called “determination through universals” based on universal mapping properties. A recently developed “heteromorphic” theory about (...) adjoints suggests a conceptual structure, albeit abstract and atemporal, for how new relatively autonomous behavior can emerge within a system obeying certain laws. The focus here is on applications in the life sciences (e.g., selectionist mechanisms) and human sciences (e.g., the generative grammar view of language). (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Toward a Radical Female Imaginary: Temporality and Embodiment in Irigaray’s EthicsEwa Plonowska Ziarek* (bio)An important intervention of Irigaray’s work on sexual difference into the postmodern debates on ethics is the mediation between two different lines of ethical inquiry: one represented by the work of Nietzsche, Deleuze, Foucault, and, to a certain degree, Castoriadis, and the other by the work of Levinas, Derrida, and Lyotard. Although the two trajectories both (...) depart from the notion of morality as a universal system of law and judgment, they represent different approaches to freedom and obligation. For Levinas, Derrida, and Lyotard, the ethical significance of alterity disrupts social systems of signification and, in this sense, marks transcendence as a break in discourse, whereas for Nietzsche, Deleuze, and Foucault, otherness is expressed within the endless variations of becoming. To elucidate this difference, it might be useful to recall John Caputo’s distinction between two kinds of postmodern ethics—the Nietzschean heteromorphism and the Levinasian heteronomy. In its emphasis on becoming and change, heteromorphism, Caputo claims, puts forward the claim of liberation but cannot accommodate an obligation to the other: “In heteromorphism, the aim is freedom from inhibition and blockage, freedom from... the weight of values... from everything that weighs the will down....” By contrast, the Levinasian ethics of alterity gives priority to the obligation to the other, which calls the freedom of the subject into question: “freedom is suspect, suspended, held in question, because it is aggressive, self-accumulative, and eventually, finally murderous. Heteronomism wants the other to be free while one is oneself held hostage” [Caputo 60].In my discussion of the radical female imaginary I would like to show that Irigaray’s work refuses to align itself with either side of this divide. Originating in the very gap between liberation and responsibility, self and other, immanence and transcendence, freedom and obligation, the feminist ethics of sexual difference cannot disregard any of these claims. For Irigaray, the ethics of sexual difference has to enable different trajectories of gendered becomings without forgetting the obligation to the other. In fact, Irigaray argues that the responsibility to the other who differs sexually is the very source of such becomings. Irigaray’s ethics maintains neither the rigid separation between freedom and obligation, as is the case in Levinas’s work, nor their symmetrical reversibility, as is the case in Kant’s ethics. By contesting the oppositions between carnal passions and ethical obligations, between the respect for the other and the becoming of the self, Irigaray allows us to think through the disjunction between the two main ethical trajectories of postmodernism and to find the means to negotiate between them. In this essay I would like [End Page 60] to focus on two moments of such negotiation: the becoming of sexed bodies and the temporality of the female imaginary.One of Irigaray’s famous claims is the charge of the forgetting of sexed bodies in the discourses of philosophy, science, linguistics, and politics. Although this claim is certainly not new in feminist theory, what is original in Irigaray’s work is the diagnosis of the numerous symptoms of such forgetting, for instance, the relation she makes between the erasure of the sexed body and the nihilism of Western culture. For Irigaray, the forgetting of sexual difference manifests itself not only in the disregard for the specificity of feminine embodiment, desires, and genealogies but also in the disembodied character of linguistic analyses, in the erasure of the drama of enunciation evident in the privilege given to predication, in the separation of the subject of knowledge from carnality and desire, in the infatuation with formalism and with its obverse side, the crippling nostalgia for the maternal body, and, finally, in the rigid separation between the immanence of flesh and the transcendence of the spirit. 1 Even more striking is the originality of Irigaray’s intervention, which links the question of sexual difference with the temporality of the body.An important element of Irigaray’s ethics of sexual difference is the recovery of “that repressed entity, the female imaginary” [TS 28]. Despite numerous excellent discussions of the imaginary in Irigaray’s work, the complexity of her negotiations around this... (shrink)

Perhaps 20% of known animal species are haplodiploid: unfertilized haploid eggs developinto males and fertilized diploid eggs into females. Sex determination in such haplodiploid species does not rely on a difference in heteromorphic sex chromosome composition but the genetic basis has been elucidated in some hymenopteran insects (wasps, sawflies, ants, bees). In these species, the development into one sex or the others depends on an initial signal whether there is only one allele or two different alleles of a single gene, (...) thecomplementary sex determiner(csd), in the zygotic genome. The gene has been most‐recently identified in the honey bee and has been found to encode an arginine serine‐rich (SR) type protein. Heterozygosity generates an active protein that initiates female development while hemizygosity/homozygosity results in a non‐active CSD protein and default male development. I will discuss plausible models of how the molecular decision of male and female is made and implemented. Comparison to hierarchies of dipteran insects suggests that SR‐type protein has facilitated the differentiation of sex‐determining systems and hierarchies. BioEssays 26:1131–1139, 2004. © 2004 Wiley Periodicals, Inc. (shrink)

Frederick E. Crowe claims that Lonergan’s thought underwent a radical transformation after the publication of Insight. In several recent articles he argues that inthe course of dealing with a problem of insight into insight and a problem of the subject as subject, Lonergan was on the verge of articulating a problem of the heteromorphism of subjectivity. I argue that Crowe’s claims depend on an uncritically selective and hermeneutically insensitive use of sources and a nest of ambiguities. By distinguishing the various (...) senses in which Lonergan uses the terms insight into and image in Insight, I show that Lonergan’s thought did not undergo the development that Crowe claims it did. A dialectical reflection on Crowe’s arguments reveals that their ambiguity arises from Crowe’s implicit adoptionof a form of cognitional atomism. (shrink)

Purpose: Sensory feedback for prosthetics is an important issue. The area of forearm stump skin that has evoked tactile sensation of fingers is defined as the projected finger map, and the area close to the PFM region that does not have ETS is defined as the non-projected finger map. Previous studies have confirmed that ETS can restore the tactile pathway of the lost finger, which was induced by stimulation of transcutaneous electrical nerve stimulation on the end of stump skin. This (...) study aims to reveal EEG features between the PFM and the NPFM regions of the stumps under the same TENS stimulation condition.Methods: The PFM and NPFM regions of the two subjects were stimulated with the same intensity of TENS, respectively. TENS as target stimuli are modulated according to the Oddball paradigm to evoke the P300 components.Result: The PFM regions of both subjects were able to elicit P300 components, while their NPFM regions were not able to elicit P300 components. However, this P300 appears early and has continuous positive peaks in front of it.Discussion: N30 and P300 can prove that the two subjects with PFM can perceive and recognize ETS. The heteromorphisms of the P300 waveform may be related to the difficulty in subjects’ cognition of ETS or caused by the fusion of P150, P200, and P300. (shrink)

Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called "internalization through a universal" based on universal mapping properties. A recently developed "heteromorphic" theory (...) of adjoint functors allows the concepts to be more easily applied empirically. This suggests a conceptual structure, albeit abstract, to model a range of selectionist mechanisms (e.g., in evolution and in the immune system). Closely related to adjoints is the notion of a "brain functor" which abstractly models structures of cognition and action (e.g., the generative grammar view of language). (shrink)

Today it would be considered "bad Platonic metaphysics" to think that among all the concrete instances of a property there could be a universal instance so that all instances had the property by virtue of participating in that concrete universal. Yet there is a mathematical theory, category theory, dating from the mid-20th century that shows how to precisely model concrete universals within the "Platonic Heaven" of mathematics. This paper, written for the philosophical logician, develops this category-theoretic treatment of concrete universals (...) along with a new concept to abstractly model the functions of a brain. (shrink)