preprint: Simplicial toposes and combinatorial homotopy

preprint: Simplicial toposes and combinatorial homotopy

[Marco Grandis]

The following preprint is available:

M. Grandis
Simplicial toposes and combinatorial homotopy,
Dip. Mat. Univ. Genova, Preprint 400 (1999).


Abstract. The term *combinatorial topos* denotes here a topos of presheaves
over a small subcategory of the category of finite sets. The main instances
we want to consider are the presheaf categories of simplicial sets, cubical
sets, and globular sets, together with their symmetric versions: e.g., the
topos  !Smp  of symmetric simplicial sets consists of all presheaves on the
category  !Delta  of finite, positive cardinals.

     We show here how combinatorial homotopy, developed in previous works
for simplicial complexes (the cartesian closed subcategory of *simple*
presheaves in  !Smp)  can be extended to the topos  !Smp.  As a crucial
advantage, the (extended) fundamental groupoid  Pi_1: !Smp --> Gpd  is left
adjoint to a natural functor  M_1: Gpd --> !Smp,  the symmetric nerve of a
groupoid, and therefore - as a strong van Kampen property - preserves all
colimits.

     Analogously, a notion of (non-reversible) *directed* homotopy can be
developed in  Smp,  with applications to image analysis similar to the ones
of the symmetric case. We have now a homotopy n-category functor  C_n: Smp
--> n-Cat,  left adjoint to a nerve  N_n = n-Cat(C_n(Delta[n]), -).  It
would be interesting to determine whether the n-category  C_n(Delta[n])
coincides with Street's oriental  O_n,  and the previous nerve with
Street's, as it seems likely.

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Available at:

ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/CmbTop.Sep99.ps

(459 K)

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Marco Grandis

Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy

e-mail: grandis@dima.unige.it

tel: +39.010.353 6805   fax: +39.010.353 6752

http://www.dima.unige.it/STAFF/GRANDIS/
ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/