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**1 - 3**of**3**### FUNDAMENTAL PUSHOUT TOPOSES

"... Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spir ..."

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Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spirit of [16, 10, 8] we replace connected components by constructively complemented, or definable, monomorphisms [1]. Unlike the locally connected case, where the fundamental groupoid is localic prodiscrete and its classifying topos is a Galois topos, in the general case our version of the fundamental groupoid is a locally discrete progroupoid and there is no intrinsic Galois theory in the sense of [19]. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case [13, 11, 7]. Throughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zero-dimensional locales [9]. In this fashion we are led to give unified and often simpler proofs of old theorems in the locally connected case, as well as new ones without that assumption.

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"... Abstract. Let S be an essentially smooth scheme over a field of characteristic exponent c. We prove that there is a canonical equivalence of motivic spectra over S MGL/(a1, a2,...)[1/c] ' HZ[1/c], where HZ is the motivic cohomology spectrum, MGL is the algebraic cobordism spectrum, and the ele ..."

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Abstract. Let S be an essentially smooth scheme over a field of characteristic exponent c. We prove that there is a canonical equivalence of motivic spectra over S MGL/(a1, a2,...)[1/c] ' HZ[1/c], where HZ is the motivic cohomology spectrum, MGL is the algebraic cobordism spectrum, and the elements an are generators of the Lazard ring. We discuss several applications including the computation of the slices of Z[1/c]-local Landweber exact motivic spectra and the convergence of the associated slice spectral sequences.