Every Elementary Higher Topos has a Natural Number Object

Abstract

We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural number object out of it. Part of the proof involves showing that various definitions of natural number objects (Lawvere, Freyd and Peano) agree with each other in an elementary $(\infty,1)$-topos. As part of this effort we also study the internal object of contractibility in $(\infty,1)$-categories, which is of independent interest. Finally, we discuss various applications of natural number objects. In particular, we use it to define internal sequential colimits in an elementary $(\infty,1)$-topos.