Aspects of the symmetric topos
Marta Bunge – 18 July 2001
This lecture will consist of a selected survey of the main (algebraic, geometric, and 2-categorical) aspects of teh symmetric monad and its generalization, the notion of an admissible KZ-doctrine [1,2,3,4,5] investigated so far.
The symmetric topos M(E) was originally constructed (using forcing methods in topos theory) as the classifier of the Lawvere distributions on a topos E bounded over a base topos S . An alternative construction was then given in  in an algebraic spirit and even having applications in algebra.
In  it is shown that the equivalence (obtained therein by other means) between S-valued distributions on E and complete spreads over E with a locally connected domain, could alternatively be seen as an equivalence between the category of points of M(E) and the category of discrete fibrations  associated with the symmetric monad on E. This approach is particularly suited to an analogy with the original construction of the spread completion given by R.H.Fox .
The algebras for the symmetric monad are identified  with the "linear objects" and this results in an interpretation of Waelbroeck's theorem in Functional Analysis  saying essentially that, for any topos E, the topos M(E) is the free "cocompletion" of E in the sense of admitting left Kan extensions along any essential geometric morphisms satisfying the BCC for comma objects.
An abstract notion of an admissible KZ-doctrine M is introduced in  as a generalization of the symmetric monad by imposing a condition on bicomma objects on the usual KZ-doctrine . In addition to the symmetric monad on TopS, there are other interesting applications, for instance in Theoretical Computer Science. In this general context, one can speak about M-fibrations and opfibrations, M-algebras, and of the M-comprehensive factorization, generalizing work of Street and Walters .
References M. Bunge, Cosheaves and distributions on toposes, Algebra Universalis 34(1995) 233-249. M. Bunge and A. Carboni, The symmetric topos, J. Pure Appl. Alg. 105(1995) 233-249. M. Bunge and J. Funk, Spreads and the symmetric topos, J. Pure Appl. Alg. 113(1996) 1-38. M. Bunge and J. Funk, Spreads and the symmetric topos II, J. Pure Appl. Alg. 130(1998) 49-84. M. Bunge and J. Funk, On a bicomma object condition for KZ-doctrines, J. Pure Appl. Alg. 143(1999) 69-105. R.H. Fox, Covering spaces with singularities, in R.H. Fox et al. (Eds.), Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz, Princeton University Press (1957) 243-257. J. Funk, The display locale of a cosheaf, Cahiers de Top. et Géo. Diff. Cat. 36(1995) 53-93. A. Kock, Monads for which structure is adjoint to units, J. Pure Appl. Alg. 104(1995) 41-59. F.W. Lawvere, Measures in Toposes, Lectures given at the workshop on Categorical Methods in Geometry, &\Acirc;arhus University, June 1983. A.M. Pitts, Lax descent for essential morphisms, Lecture, Cambridge Categories Conference, July 1986. R. Street, Fibrations in bicategories, Cahiers de Top. et Géo Diff. Cat. 21(1980) 111-160. R. Street and R.F.C. Walters, The comprehensive factorization of a functor, Bull. Amer. Math. Soc. 79(1973) 936-941. L. Waelbroeck, Differentiable mappings into b-spaces, J. Funct. Analysis 79(1973) 936-941.