Given an augmented simplicial object $d\_\bullet:X\_\bullet \to \Delta X\_{-1}$, suppose there's a simplicial map $s\_\bullet :\Delta X\_{-1}\to X\_\bullet$ making $d\_\bullet$ a deformation retract, i.e such that $d\_\bullet$ is both a retract and a homotopy-section of $s\_\bullet$. This is equivalent to providing the augmented simplicial object with an "extra degeneracy" (just an alternative description of the simplicial homotopy axioms in this case). Such an extra degeneracy of an augmented simplicial object will be called a *splitting*.


Let $\mathrm{S}$-$s\mathsf C$ denote the category of split simplicial objects (with fixed splitting) and simplicial arrows between them respecting the simplicial homotopies. This category admits a forgetful functor to simplicial objects $s\mathsf C$.


On [page 20](https://books.google.com.au/books?id=427jnAAACAAJ&lpg=PP1&pg=PA20#v=onepage&q=20&f=false) of Duskin's *Simplicial Methods and the Interpretation of Triple Cohomology* ([AMS page](http://bookstore.ams.org/memo-3-163/)), the author remarks this forgetful functor is a *left* adjoint to a shifting functor defined by deleting the top face map, viewing the top degeneracy as an extra one, and shifting to a lower index. (The simplicial homotopy is defined by $h\_i=s\_0^{n-i}s\_{n+1}d\_0^{n-i}$.)


Moreover, the author writes this shifting functor $s\mathsf C\to \mathrm{S}$-$s\mathsf C$ is monadic. In other words, simplicial objects are monadic over split simplicial objects.


1. What's the intuition behind the fact the shifting functor actually takes values in split simplicial objects? This seems strange to me - as if saying a simplicial object becomes contractible if you forget the top face map. How can that be?
2. What's the intuition behind monadicity? An algebra over a split simplicial object (which is already a structured simplicial object) is an arrow to it, so how can additional structure on an already structured object yield back the original notion of object and arrow?


**Added.** I am looking for naive *geometric* intuition for these facts, namely: 1. that a simplicial object becomes contractible upon merely forgetting a face map and reindexing; 2. simplicial objects are monadic over split ones. Ideally, I would like an example of what the contraction deformation retract actually does to the Décalage of a non-contractible simplicial complex, say the boundary of a tetrahedron.



Why are simplicial objects monadic over split (contractible) simplicial objects?
Coming across this old question and noticing it still unanswered, here’s an attempt at the “geometric” answer OP wanted.


First note that a split simplicial complex isn’t in general contractible — it’s just *1-truncated*, i.e. each *connected component* is contractible. Concretely, the augmentation indexes the connected components, and the splitting + homotopy gives a distinguished point + contraction for each component.


Write $R : \mathrm{SSet} \to \mathrm{SplSSet}$ for the desired right adjoint. We have $(RX)\_{-1} = X\_0$, so $X\_0$ indexes the components of $RX$, and then for each $x \in X\_0$, the $n$-simplices of $RX$ in the component over $x$ are the $n+1$-simplices whose $(n+1)$th vertex is $x$ (or $0$th vertex, depending on your indexing convention). In other words the $x$-component of $RX$ is the “out-star” (or “out-cocone”, or “out-path-space”) of $x$, consisting of all simplices whose *source* is $x$. But now each out-star is contractible, contracting into the degenerate 1-simplex on $x$; this is easy to check algebraically, but also easy to see geometrically/topologically — the space of “paths in $X$ with source $x$” is contractible to the constant path on $x$.


Briefly: **Under the right adjoint, a simplicial set falls apart into the collection of out-stars of its vertices; and each out-star is clearly contractible to the reflexivity path on that vertex.**


