# On the construction of the simplicial category $\Delta$

Is a classical example that in the topos $Set$ the set of natural numbers (finite cardinals) $\mathbb{N}$ is the natural-numbers objet as in topos theory definition. Now the category $\Delta$ has for objects the natural numebers ($n\in \mathbb{N}$ as finite cardinals is the set $\lbrace0,\ldots, n-1\rbrace$) and for morphisms the ordered maps .

I ask if this is this is generalizable for a general topos: give a topos $\mathcal{E}$ with a natural number object $\mathbb{N}$, build from $\mathbb{N}$ a construction of a internal category $\Delta$ (with object of objects $\mathbb{N}$) that generalize the usual above.

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Perhaps it is a good idea to follow the universal property of $\Delta$ (relative to the topos $\mathrm{Set}$), which says that it is the universal strict monoidal category together with a chosen monoid object. –  Martin Brandenburg May 26 '12 at 16:46
It is certainly possible to perform such a construction and obtain an internal category in $\mathcal{E}$. Is that all you are asking? –  Mike Shulman May 26 '12 at 19:46
Shulman: Yes, I wish know how get the internal category $\Delta$ form $\mathbb{N}$ in terms of internal constructions inside a topos. I think that $\Delta$ is a very foundamental and natural structure as $\mathbb{N}$, then I guess exist a elementar (in terms of internal categorical language) way to get the former from the letter. –  Buschi Sergio May 26 '12 at 20:07

The construction I think you are looking for is due to Benabou and given at the end of the paper "Some remarks on free monoids in a topos", LNM1488, Category theory (Como, 1990), 20–29. Here is a summary of how it goes.

Given a functor $U:A \to B$ with left adjoint $F$ one obtains for each $X \in A$ a simplicial object in $A$ with object of 0-simplices $FUX$, object of 1-simplices $FUFUX$ and face and degeneracy maps built from the multiplication and unit for the monad $T=UF$. If the adjunction is monadic and the monad is cartesian, meaning that $T$ preserves pullbacks and the naturality squares of the unit and multiplication are pullback squares then that simplicial object is in fact an internal category in $A$ with object of objects $FUX$. Let us call it $QX$.

For example consider the forgetful functor $U:Mon \to Set$ from the category of monoids to the category of sets. The free monoid monad is cartesian, so that associated to each monoid $X \in Mon$ you have a category internal to monoids, a strict monoidal category $QX$. If you take the terminal monoid $1$ then $FU1$ is the monoid of natural numbers $\mathbf{N}$ and you can check directly that the strict monoidal category $Q1$ is $\Delta$ with the usual ordinal sum.

Benabou does everything in the context you ask for - in a topos E with a natural numbers objects. He proves, for instance, that monoids in E are then monadic over E, with the monoid monad cartesian, so that you can associate a strict monoidal category to any monoid in E.

I don't think a natural numbers object in a topos need be of the form FUX for a monoid in E - I'm not sure as I don't know much topos theory. You would need the natural numbers object to be of this form for the above construction to give an internal category in Mon(E) with object of objects the natural numbers object.

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Is a at pag. 28-29 of this article, thank –  Buschi Sergio May 27 '12 at 16:12
But this really constructs $\Delta_+$ since you need the empty ordinal to have a monoidal unit. –  David Carchedi May 28 '12 at 11:14