Re: a question

Re: a question

[Marco Grandis]

On 18 Nov 2015, at 14:51, Vladimir Voevodsky wrote:

> Hello,
> could anyone point me out to a proof that the category F whose
> objects are natural numbers and morphisms are morphisms of finite
> sets is a free category with coproducts where the associativity and
> unity isomorphisms are identities generated by one object?
> Thank you in advance,
> Vladimir.

In this paper
M. Grandis, Finite sets and symmetric simplicial sets, Theory Appl.
Categ. 8 (2001), No. 8, 244-252
the following results are proved, and point (a') should be related to
what you want:

The category of finite cardinals, equivalent to the category of
finite sets and the site of augmented symmetric simplicial sets, is:
(a') the free strict monoidal category with an assigned symmetric
monoid;
(b') the subcategory of Set generated by finite cardinals, their
faces, degeneracies and main transpositions;
(c') the category generated by faces, degeneracies and main
transpositions, under the symmetric cosimplicial relations.

The properties above are related to well-known (see Mac Lane)
characterisations of the category of finite ordinals, the site of
augmented simplicial sets:
(a) the free strict monoidal category with an assigned internal monoid;
(b) the subcategory of Set generated by finite ordinals, their faces
and degeneracies;
(c) the category generated by faces and degeneracies, under the
cosimplicial relations.

Regards,   Marco


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a question

[Vladimir Voevodsky]

Hello,

could anyone point me out to a proof that the category F whose objects are natural numbers and morphisms are morphisms of finite sets is a free category with coproducts where the associativity and unity isomorphisms are identities generated by one object?

Thank you in advance,
Vladimir.


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a question

[André Joyal]

Dear all,

The following question was privately raised by one of my correspondant:

  >In Set we have a large (but exponentiable) canonical category of  
finite sets.
  >Question: What is the relation of (any) of the small (= internal)
  >categories S_f with the notions of finite object in a topos ?.

Let me try to answer it:

The internal stack of finite sets S_f is classifying finite objects
exactly like the Lawvere object is classifying monomorphisms.

Is there another answer?

Best,
André




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