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**1 - 3**of**3**### Is Chemistry a Branch of

- Physics’, Journal for General Philosophy of Science
, 1982

"... coverings of quasi locally connected toposes ..."

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### 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.

### QUASI LOCALLY CONNECTED TOPOSES

"... Abstract. We have shown [2, 4] that complete spreads (with a locally connected domain) over a bounded topos E (relative to S) are ‘comprehensive ’ in the sense that they are precisely the second factor of a factorization associated with an instance of the comprehension scheme [8, 12] involving S-val ..."

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Abstract. We have shown [2, 4] that complete spreads (with a locally connected domain) over a bounded topos E (relative to S) are ‘comprehensive ’ in the sense that they are precisely the second factor of a factorization associated with an instance of the comprehension scheme [8, 12] involving S-valued distributions on E [9, 10]. Lawvere has asked whether the ‘Michael coverings ’ (or complete spreads with a definable dominance domain [3]) are comprehensive in a similar fashion. We give here a positive answer to this question. In order to deal effectively with the comprehension scheme in this context, we introduce a notion of an ‘extensive topos doctrine, ’ where the extensive quantities (or distributions) have values in a suitable subcategory of what we call ‘locally discrete’ locales. In the process we define what we mean by a quasi locally connected topos, a notion that we feel may be of interest in its own right.