Contents

topos theory

# Contents

## Idea

The Schanuel topos (also called the Myhill-Schanuel topos) is the Grothendieck topos of combinatorial functors. It plays an important role in computer science in the theory of name-binding calculi and in William Lawvere‘s approach to petit toposes. It can be viewed as a categorical variant of the Fraenkel-Mostowski model of set theory.

## Definition

The Schanuel topos is the sheaf topos $Sh(FinSet^{op}_{mono},J)$ where $FinSet^{op}_{mono}$ is the opposite of the category of finite sets and monomorphisms and the coverage is the collection of all sieves generated by singletons $\{f\}\quad.$

## Properties

• The objects of the Schanuel topos are called nominal sets and correspond precisely to the pullback preserving functors $FinSet_{mono}\to Set$.

• The Schanuel topos $Sh(FinSet^{op}_{mono},J)$ is atomic over $Set$ (S. Schanuel, cf. (Wraith 1978), p.335) hence Boolean. This fact can be viewed as a reflex of the urelements in Fraenkel-Mostowski set theory.

• $Sh(FinSet^{op}_{mono},J)$ is the classifying topos $Set[D_\infty]$ for the theory of infinite decidable objects $D_\infty$ i.e. for a Grothendieck topos $\mathcal{E}$ geometric morphisms $\mathcal{E}\to Sh(FinSet^{op}_{mono},J)$ correspond to infinite decidable objects in $\mathcal{E}$. $Sh(FinSet^{op}_{mono},J)$ is equivalent to $Sh_{\neg\neg}([FinSet_mono,Set])$.

• $Sh(FinSet^{op}_{mono},J)$ is the category of continuous actions for the group of bijections of $N$ equipped with the topology derived from the product topology for $\prod_{N}N\quad .$

• $Sh(FinSet^{op}_{mono},J)$ is the Kleisli category of the monad on the topos of species $Set^{FinSet_{iso}}$ induced by the inclusion of finite sets and bijections $FinSet_{iso}\hookrightarrow FinSet_{mono}$ (cf. Fiore-Menni 2004).

### Remark

For some information on the history of the Schanuel topos see section 10 of Menni (2009, pp.529f).

• M. Fiore, M. Menni, Reflective Kleisli Subcategories of the Category of Eilenberg-Moore Algebras for Factorization Monads , TAC 15 no. 2 (2004) pp.40-65. (pdf)

• M. J. Gabbay, A. M. Pitts, A new approach to abstract syntax with variable binding, Formal Aspects of Computing 13 (2002) pp.341-363. (draft)

• Peter Johnstone, Sketches of an Elephant vols. I,II, Oxford UP 2002. (pp.79f, 691, 925)

• F. William Lawvere, Qualitative Distinctions between some Toposes of Generalized Graphs , Cont. Math. 92 (1989). (pp.298f)

• S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994. (pp.155, 158)

• Matías Menni, About N-quantifiers , Appl. Cat. Struc. 11 (2003) pp.421-445. (preprint)

• Matías Menni, Algebraic categories whose projectives are explicitly free , TAC 22 no.20 (2009) pp.509-541. (abstract)

• Sam Staton, Name-passing process calculi: operational models and structural operational semantics, Technical Report 688 CL University Cambridge 2007. (pdf)

• G. Wraith, Intuitionistic Algebra: Some Recent Developments of Topos Theory , Proc. ICM Helsinki (1978) pp.331-337. (pdf)