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On theories of superalgebras of differentiable functions

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David Carchedi and Dmitry Roytenberg

This is the first in a series of papers laying the foundations for a
differential graded approach to derived differential geometry (and other
geometries in characteristic zero). In this paper, we study theories of
supercommutative algebras for which infinitely differentiable functions
can be evaluated on elements. Such a theory is called a *super Fermat
theory.* Any category of superspaces and smooth functions has an
associated such theory. This includes both real and complex
supermanifolds, as well as algebraic superschemes. In particular, there is
a super Fermat theory of *$C^\infty$-superalgebras.*
$C^\infty$-superalgebras are the appropriate notion of supercommutative
algebras in the world of $C^\infty$-rings, the latter being of central
importance both to synthetic differential geometry and to all existing
models of derived smooth manifolds. A super Fermat theory is a natural
generalization of the concept of a Fermat theory introduced by E. Dubuc
and A. Kock. We show that any Fermat theory admits a canonical
*superization,* however not every super Fermat theory arises in this
way. For a fixed super Fermat theory, we go on to study a special
subcategory of algebras called near-point determined algebras, and derive
many of their algebraic properties.

Keywords:
$C^\infty$-ring, Lawvere theory, superalgebra, supergeometry

2010 MSC:
Primary: 18C10, 58A03 ; Secondary: 58A50, 17A70

*Theory and Applications of Categories,*
Vol. 28, 2013,
No. 30, pp 1022-1098.

Published 2013-10-24.

http://www.tac.mta.ca/tac/volumes/28/30/28-30.dvi

http://www.tac.mta.ca/tac/volumes/28/30/28-30.ps

http://www.tac.mta.ca/tac/volumes/28/30/28-30.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/28/30/28-30.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/28/30/28-30.ps

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