The category of toric stacks

In this paper, we show that there is an equivalence between the 2-category of smooth Deligne–Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine–Illusie plays a central role in obtaining our results.

A conceptuocentric shift in the characterization of language

Recognizing limitations of the “syntactocentric” perspective, Jackendoff proposes a model in which phonology, syntax, and conceptual systems are each independently combinatorial. We can ask, however, whether he has taken this issue to its logical conclusion. The fundamental question that is not fully addressed is whether the combinatorial aspect of syntax originated in, and derives from, the indeed “far richer” conceptual system, a question to be discussed.