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Starting with a simplicial set $K$ one can view it as a semi-simplicial set, then produce a simplicial category. … I would like to know under which conditions on $K$ this simplicial category is equivalent to the simplicial category which one gets by applying the usual functor from simplicial sets to simplicial categories …

Given two simplicial topological spaces $X_{\bullet}$ and $Y_{\bullet}$ (i.e. a simplicial object in Top) and a continuous map between their geometric realizations $f \colon \lvert X_{\bullet} \rvert \ … Is $f$ homotopic to $\lvert \varphi_{\bullet} \rvert$ for a map $\varphi_{\bullet}$ of simplicial spaces? …

Clarification: A simplicial functor between simplicial model categories $M$ and $N$ is a simplicial enriched functor between simplicial enriched categories. … on $M$ and on simplicial sets. …

I restricted the question from arbitrary diagrams to simplicial diagrams. … Suppose we have a simplicial object in finite CW complexes. …

I begin by recalling the definition of contiguous simplicial maps between abstract simplicial complexes: Definition. … Has it been studied a notion of contiguity for simplicial maps between simplicial sets (or $\Delta$-complexes)? If that is the case could you provide me a reference. Thanks in advance. …

(or replace) the theory of simplicial spaces with a theory of internal locales to $\mathbf{sSet}$ or something similar. … Can you create a constructive version of classical Hodge theory, such that internalized to simplicial sets we retrieve the theory of mixed Hodge theory? …

$\DeclareMathOperator\Sing{Sing}$I'm writing a paper about simplicial sets and how they may “replace” simplicial complexes in some known results. … Let $K$ be a simplicial complex with vertex set $V$, and choose an ordering on $V$. …

I recently was wondering if there was a name for sheaves which were locally constant on the open simplexes in a simplicial complex. After some googling I stumbled across simplicial sheaves. … Are simplicial sheaves related to the locally-constant-on-simplex etale sheaves of a simplicial complex? If not, does this concept have a different name? …

My questions are: 1) Are there any natural maps between a simplicial space and its simplicial replacement? … 2) Why is the homotopy colimit of a simplicial space weakly equivalent to its geometric realization? …

How should one think about simplicial objects in a category versus actual objects in that category? … For example, both for intuition and for practical purposes, what's the difference between a [commutative] ring and a simplicial [commutative] ring? …

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. … In other words, simplicial objects are monadic over split simplicial objects. What's the intuition behind the fact the shifting functor actually takes values in split simplicial objects? …

Simplicial posets are generalizations of simplicial complexes (see, e.g., http://math.mit.edu/~rstan/pubs/pubfiles/82.pdf). … Have these matroidal simplicial posets been studied at all? Are there conjectures (e.g., about face numbers) for them? …

Given an abstract simplicial complex $K$ on a set of vertices $V$, we can form a semi-simplicial set by $F(K)$ sending $F(K)_n$ to be the set of ordered $(n+1)$-tuples of vertices in $V$ forming an $n$ … One can add in degeneracies by hand to make this a simplicial set. In general, $|F(K)|$ is not homotopy equivalent to $K$; for example, $|F(\Delta^n)|$ is a wedge of $n$-spheres. …

A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. … that is, is every simplicial set, having a finite number of simplicies in each degree, necessarily finite? …

There is the notation of a geometric realization of a finite abstract simplicial complex: Let $D=(S,D)$ be a finite abstract simplicial complex. Then choose a total order on $S$, w.l.o.g. … If I am not mistaken there are finite topological simplicial complexes which are not the geometric realization of a finite abstract simplicial complex. …