A model of type theory in simplicial sets: A brief introduction to Voevodsky's homotopy type theory

We describe how to interpret constructive type theory in the topos of simplicial sets where types appear as Kan complexes and families of types as Kan fibrations. Since Kan complexes may be understood as weak higher-dimensional groupoids this model generalizes and extends the (ordinary) groupoid model which was introduced by M. Hofmann and the author about 20 years ago. Finally, we discuss Voevodsky's Univalence Axiom which has been shown to hold in this model. This axiom roughly states that isomorphic types are equal. The type theoretic notion of isomorphism provided by this model coincides with homotopy equivalence of Kan complexes. For this reason it has become common to refer to it as Homotopy Type Theory.     

Homotopy and heterotopy and the bilateral field advantage in the Dimond paradigm

Pairs of homotopic and heterotopic visual bilateral stimuli and pairs of unilateral visual stimuli were presented to 12 normal right-handed university students requiring a key press if they were of the same form. As predicted from the known histology of the corpus callosum (massive preponderance of homotopic fibers), homotopic presentations yielded significantly faster reaction times than heterotopic stimulations. Bilateral pairs of stimuli were also advantaged in comparison with unilateral trials, replicating Sereno and Kosslyn [Sereno, A. B., & Kosslyn, S. M. (1991). Discrimination within and between hemifields: a new constraint on theories of attention. Neuropsychologia, 29, 659-675]. Moreover, certain attentional processes have never been investigated in the Dimond paradigm and this study provides evidence to the effect that discriminative reaction times to stimulus pairs are strongly influenced by their proximity to the fixation point. In similar previous experiments, the homotopy/heterotopy observation and the bilateral field advantage may have been distorted by that particular confound, as well as several others.