Abstract
We give a number of equivalent conditions for a topos to be homotopically trivial and then relate these conditions to the logic of the topos. This is accomplished by constructing a family of intervals that can detect complemented, regular subobjects of the terminals. It follows that these conditions generally are weaker than the Stone condition but are equivalent to it if they hold locally. As a consequence we obtain an extension of Johnstone’s list of conditions equivalent to DeMorgan’s law. Thus, for example, the fact that there is no nontrivial homotopy theory in the category of sets is equivalent to the fact, among others, that maximal ideals in commutative rings are prime. Moreover, any topos has a ‘best approximation’ by a locally homotopically trivial topos.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 171-182 |
| Number of pages | 12 |
| Journal | Pacific Journal of Mathematics |
| Volume | 110 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jan 1 1984 |
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ASJC Scopus subject areas
- Mathematics(all)
Cite this
Homotopically trivial toposes. / Mielke, Marvin.
In: Pacific Journal of Mathematics, Vol. 110, No. 1, 01.01.1984, p. 171-182.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Homotopically trivial toposes
AU - Mielke, Marvin
PY - 1984/1/1
Y1 - 1984/1/1
N2 - We give a number of equivalent conditions for a topos to be homotopically trivial and then relate these conditions to the logic of the topos. This is accomplished by constructing a family of intervals that can detect complemented, regular subobjects of the terminals. It follows that these conditions generally are weaker than the Stone condition but are equivalent to it if they hold locally. As a consequence we obtain an extension of Johnstone’s list of conditions equivalent to DeMorgan’s law. Thus, for example, the fact that there is no nontrivial homotopy theory in the category of sets is equivalent to the fact, among others, that maximal ideals in commutative rings are prime. Moreover, any topos has a ‘best approximation’ by a locally homotopically trivial topos.
AB - We give a number of equivalent conditions for a topos to be homotopically trivial and then relate these conditions to the logic of the topos. This is accomplished by constructing a family of intervals that can detect complemented, regular subobjects of the terminals. It follows that these conditions generally are weaker than the Stone condition but are equivalent to it if they hold locally. As a consequence we obtain an extension of Johnstone’s list of conditions equivalent to DeMorgan’s law. Thus, for example, the fact that there is no nontrivial homotopy theory in the category of sets is equivalent to the fact, among others, that maximal ideals in commutative rings are prime. Moreover, any topos has a ‘best approximation’ by a locally homotopically trivial topos.
UR - http://www.scopus.com/inward/record.url?scp=84972584469&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84972584469&partnerID=8YFLogxK
U2 - 10.2140/pjm.1984.110.171
DO - 10.2140/pjm.1984.110.171
M3 - Article
AN - SCOPUS:84972584469
VL - 110
SP - 171
EP - 182
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
SN - 0030-8730
IS - 1
ER -