The principal fibration sequence and the second cohomotopy set

Let $p:E -> B$ be a principal fibration with classifying map $w:B -> C$. It is well-known that the group $[X,\Omega C]$ acts on $[X,E]$ with orbit space the image of p_#, where p_#: [X,E] -> [X,B]. The isotropy subgroup of the map of $X$ to the base point of $E$ is also well-known to be the image of $[X, \Omega B]$. The isotropy subgroups for other maps $e:X -> E$ can definitely change as $e$ does. The set of homotopy classes of lifts of $f$ to the free loop space on $B$ is a group. If $f$ has a lift to $E$, the set p_#^{-1}(f) is identified with the cokernel of a natural homomorphism from this group of lifts to $[X, \Omega C]$. As an example, $[X,S^2]$ is enumerated for $X$ a 4-complex.Comment: 12 page

Languages adapt to their contextual niche

abstractIt is well established that context plays a fundamental role in how we learn and use language. Here we explore how context links short-term language use with the long-term emergence of different types of language system. Using an iterated learning model of cultural transmission, the current study experimentally investigates the role of the communicative situation in which an utterance is produced (situational context) and how it influences the emergence of three types of linguistic systems: underspecified languages (where only some dimensions of meaning are encoded linguistically), holistic systems (lacking systematic structure), and systematic languages (consisting of compound signals encoding both category-level and individuating dimensions of meaning). To do this, we set up a discrimination task in a communication game and manipulated whether the feature dimension shape was relevant or not in discriminating between two referents. The experimental languages gradually evolved to encode information relevant to the task of achieving communicative success, given the situational context in which they are learned and used, resulting in the emergence of different linguistic systems. These results suggest language systems adapt to their contextual niche over iterated learning.</jats:p