The notion of a natural model of type theory is defined in terms of that of a representable
natural transfomation of presheaves. It is shown that such models agree exactly with the …

Any scheme has its associated little and big Zariski toposes. These toposes support an
internal mathematical language which closely resembles the usual formal language of …

We define an elementary $\infty $-topos that simultaneously generalizes an elementary
topos and Grothendieck $\infty $-topos. We then prove it satisfies the expected topos …

There are so many different notions of “space”(topological spaces, manifolds, schemes,
stacks, and so on, as discussed in various other chapters of this book and its companion …

We present XTT, a version of Cartesian cubical type theory specialized for Bishop setsa la
Coquand, in which every type enjoys a definitional version of the uniqueness of identity …

Cheng, Gurski, and Riehl constructed a cyclic double multicategory of multivariable
adjunctions. We show that the same information is carried by a double polycategory, in …

We study elementary theories of well-pointed toposes and pretoposes, regarded as category-
theoretic or “structural” set theories in the spirit of Lawvere's “Elementary Theory of the …

arXiv:1703.03007v1 [math.CT] 8 Mar 2017 Page 1 arXiv:1703.03007v1 [math.CT] 8 Mar 2017
Homotopy type theory: the logic of space Michael Shulman∗ March 10, 2017 1 Introduction to …

This is the author's PhD thesis. It is a contribution to categorical logic, in particular to the
theory of realizability toposes. While the tools of categorical logic have proven very …

We extend the usual internal logic of a (pre) topos to a more general interpretation, called
the stack semantics, which allows for" unbounded" quantifiers ranging over the class of …