In several works (es. [CS]) the study of the properties of the simplicial category $\Delta$ reveals fundamental aspects of universal properties (eg monoid generator) or basic constructions (eg decalage construction, see also [D]). A far as I know, however, the following fact (which I hope is true and not wrong as often happens to me) has never been detected:

I define (or build) a simplicial object in the same $ \Delta $, ie a functor $ D: \Delta^{op} \to \Delta $, this allows (by restriction or by Kan extension) to associate a co.simplicial object from a simplicial one (in any category) and vice versa.

Let $\Delta$ the simplicial category (without the empty set $\emptyset$), as usual given a category $\mathcal{C}$ a functor $X: \Delta^{op}\to \mathcal{C}$ is called a simplicial object of $\mathcal{C}$.

I show that exist a "immersion" simplicial object in $\Delta$ itself i.e. a functor $D: \Delta^{op}\to \Delta$ injective on objets and morphisms. On objects define $D([n]):=[n+1]$ for $n=0,1,\ldots$, and for $\sigma_n^i: [n+1]\to [n], 0\leq i\leq n$, $\partial_n^i: [n-1]\to [n], 0\leq i\leq n$ let $D(\sigma_{n-1}^i):= \partial_n^{i+1}$ and $D(\partial_n^i):= \sigma_n^i$.

In intuitive way: $\sigma^i$ "overlaps at $j$", then associate to it the map $D(\sigma^j)=\partial^{j+1}$ that "split at $j$".

As usual let $D_n:= D([n])=[n+1]$, $s_i:= D(\sigma^i)=\partial^{i+1}$, $d_i:= D(\partial^i)=\sigma^i$, for proving thet $D$ is a functor I check the following three point (see [GZ] p. 24-25):

1) $d_i d_j= d_{j-1} d_i\ i<j$ i.e. $\sigma^i \sigma^j= \sigma^{j-1} \sigma^i\ i<j$ i.e (posing $j':= j-1$) $\sigma^i \sigma^{j'+1}= \sigma^{j'} \sigma^i\ i\leq j'+2$ and this last is true.

2) $s_is_j=s_{j+1} s_i\ i\leq j$ i.e. $\partial^{i+1} \partial^{j+1}=\partial^{j+2} \partial^{i+1}\ i\leq j$ i.e. $\partial^{i} \partial^{j}=\partial^{j+1} \partial^{i}\ 0<i\leq j$ i.e (posing $j':= j+1$) $\partial^{i} \partial^{j'-1}=\partial^{j'} \partial^{i}\ 0<i<j'$ and this is true.

3) this is in tree part:

3.1) $d_is_j=s_{j-1}d_i\ i<j$ i.e. $\sigma^i\partial^{j+1}=\partial^j\sigma^i\ i<j$ i.e. $\sigma^i\partial^{j+1}=\partial^j\sigma^i\ i+1<j+1$ i.e. (posing j'=j+1) $\sigma^i\partial^{j'}=\partial^{j'-1}\sigma^i\ i+1<j'$ i.e. (changing index) $\sigma^j\partial^i=\partial^{i-1}\sigma^j\ j+1<i$ that is true.

3.2) $d_i s_i=d_{i+1} s_i=1$ i.e. $\sigma^i\partial^{i+1}=\partial^{i+1}\sigma^{i+1}=1$ that is true.

3.3) $d_i s_j=s_j d_{i-1}\ i>j+1$ i.e. $\sigma^i \partial^{j+1}=\partial^{j+1} \sigma^{i-1}\ i>j+1$ i.e. (posing $j'=j+1$) $\sigma^i \partial^{j'}=\partial^{j'} \sigma^{i-1}\ i>j'$ i.e. (changing index) $\sigma^j \partial^i=\partial^i \sigma^{j-1}\ i<j$ that is true.

I ask if this fact is just knowed in Mathematical literature, and if this can be useful in the mathematical sectors involving simplical theory.

[D] J.Duskin "Simplicial methods and the interpretation of "triple" cohomology, number 163 in Mem Amer. Math. "

[CS]D. Verity "complicial Sets" see http://arxiv.org/abs/math/0410412

[GZ] P. Gabriel, M. Zisman "Calculus of Fractions and Homotopy theory" Springer.

  • 6
    $\begingroup$ One place where this has been published before is in section 3.11 of Ross Street, Fibrations in bicategories, Cahiers 21(2), 1980. $\endgroup$ – Finn Lawler May 15 '14 at 12:08

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