A General Framework for the Semantics of Type Theory

We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin-L\"{o}f type theory, two-level type theory and cubical type theory. We establish basic results in the semantics of type theory: every type theory has a bi-initial model; every model of a type theory has its internal language; the category of theories over a type theory is bi-equivalent to a full sub-2-category of the 2-category of models of the type theory

Superhump-like variation during the anomalous state of SU UMa

We observed an anomalously outbursting state of SU UMa which occurred in 1992. Time-resolved photometry revealed the presence of signals with a period of 0.0832 +/- 0.0019 d, which is 3.6 sigma longer than the orbital period (0.07635 d) of this system. We attributed this signal to superhumps, based on its deviation from the orbital period and its characteristic profile. During this anomalous state of SU UMa, normal outbursts were almost suppressed, in spite of relatively regular occurrences of superoutbursts. We consider that an ensuing tidally unstable state following the preceding superoutburst can be a viable mechanism to effectively suppress normal outbursts, resulting in an anomalously outbursting state.Comment: 3 pages, 4 figures, accepted for publication in Astronomy and Astrophysic

Fibred Fibration Categories

We introduce fibred type-theoretic fibration categories which are fibred categories between categorical models of Martin-L\"{o}f type theory. Fibred type-theoretic fibration categories give a categorical description of logical predicates for identity types. As an application, we show a relational parametricity result for homotopy type theory. As a corollary, it follows that every closed term of type of polymorphic endofunctions on a loop space is homotopic to some iterated concatenation of a loop

Theory of Interface: Category Theory, Directed Networks and Evolution of Biological Networks

Biological networks have two modes. The first mode is static: a network is a passage on which something flows. The second mode is dynamic: a network is a pattern constructed by gluing functions of entities constituting the network. In this paper, first we discuss that these two modes can be associated with the category theoretic duality (adjunction) and derive a natural network structure (a path notion) for each mode by appealing to the category theoretic universality. The path notion corresponding to the static mode is just the usual directed path. The path notion for the dynamic mode is called lateral path which is the alternating path considered on the set of arcs. Their general functionalities in a network are transport and coherence, respectively. Second, we introduce a betweenness centrality of arcs for each mode and see how the two modes are embedded in various real biological network data. We find that there is a trade-off relationship between the two centralities: if the value of one is large then the value of the other is small. This can be seen as a kind of division of labor in a network into transport on the network and coherence of the network. Finally, we propose an optimization model of networks based on a quality function involving intensities of the two modes in order to see how networks with the above trade-off relationship can emerge through evolution. We show that the trade-off relationship can be observed in the evolved networks only when the dynamic mode is dominant in the quality function by numerical simulations. We also show that the evolved networks have features qualitatively similar to real biological networks by standard complex network analysis.Comment: 59 pages, minor corrections from v