By Ieke Moerdijk, J. J. Vermeulen

We advance the speculation of compactness of maps among toposes, including linked notions of separatedness. This idea is outfitted round types of 'propriety' for topos maps, brought right here in a parallel model. the 1st, giving what we easily name 'proper' maps, is a comparatively susceptible situation as a result of Johnstone. the second one form of right maps, the following known as 'tidy', fulfill a better as a result of Tierney and Lindgren.Various different types of the Beck-Chevalley situation for (lax) fibered product squares of toposes play a principal position within the improvement of the speculation. functions comprise a model of the Reeb balance theorem for toposes, a characterization of hyperconnected Hausdorff toposes as classifying toposes of compact teams, and of strongly Hausdorff coherent toposes as classifiying toposes of profinite groupoids. Our effects additionally permit us to strengthen extra specific points of the factorization conception of geometric morphisms studied via Johnstone. Our ultimate software is a (so-called lax) descent theorem for tidy maps among toposes. This theorem implies the lax descent theorem for coherent toposes, conjectured by way of Makkai and proved past via Zawadowski.

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6). Let Let m- fr,+' n dcl / v '■*r 1I «? « « s == EI(v ( 'r +! 13) C h a p t e r 3. R a n d o m i z e d e x t e n d e d s t o c h a s t i c integrals w i t h j u m p s 40 where Elr = {0 = r 0 < rj < ■ • ■ < r m = 1} is a partition of [0,1]. Then u G £2'> n I 4 ([0,1] x n ) , u n - t w in the norm Eq. 6) as max |Ar,| -> 0, and it follows from Eq. 12) that M £ Q1ust,(Atk,s)dWs'j w u t=0 2 n-1 -YJ(Kj\^n{Mk,s)dWi ' 0 k=0 as m a x | A r , | —> 0 uniformly with respect to fl ( . 14) k=0 KJ ° J ' ° 0

T) drds J 2 1 <\\u\\l\\N\\l(supMj \DrVt\Ur) \D o T VjVrj <\H\ |yv|| (supMy < < ++oo. oo. 32) Chapter 3. Randomized extended stochastic integrals with jumps 52 But Eq. 32) may be proved in a similar way as Eq. 31). Thus we established Eq. 28). Vt\ \fi\(dt,s)ds<\\u\\2 supM|DsV;N < oo (0,T) then the second term in Eq. 27) converges to zero. Vt\ —> 0. Hence the first term in Eq. 33) converges to zero in L1(S7). Consequently S2—> J \ . J $lx(Vf,Ut)DsVtii{dt,s)ds [0,T) in probability. 4. Xou. T) follows from the continuity of $ " Vt, Ut, the inequality M f N(s) | « , | / ' | C s £ r |