Are there tetrahedra which can be subdivided into three parts similar to the original? I believe this would require splitting one face into three parts. I know some types of tetrah …

Is there a known definition of vector fields on a simplicial manifold? For me, it seems natural that the definition should be something along the lines: Let $M_{\bullet}$ be a si …

An edge of a triangulated manifold is said to be contractible if it may be contracted to a vertex without modifying the topological type of the underlying manifold. Otherwise, the …

In the paper "Homotopy Lie algebras", Schechtman and Hinich proved that any cosimplicial differential graded Lie algebra has the structure of a 'Lie May algebra'. If my understan …

Consider a site S (I am mostly interested in hypercomplete sites, e.g., the site of smooth manifolds). The category of simplicial presheaves SPSh(S) on S can be equipped with the l …

Let $A, X$ be simplicial sets, and suppose there's an inclusion $A \longrightarrow X$. Geometrically realizing the inclusion map, we get a pair of spaces $(\mathcal{A}, \mathcal{X …

Let $K_1$ and $K_2$ be two simplicial complexes. I am seeking a measure of the distance between $K_1$ and $K_2$ when viewed as combinatorial objects. What I have in mind is somethi …

Question Are there efficient algorithms to check if a finite simplicial complex defined in terms of its maximal facets is shellable? By efficient here I am willing to consid …

Let $S$ be a (small) symmetric monoidal category and $X$ a (small) category on which $S$ acts. $\pi_0(S) = \pi_0(BS)$ is naturally an abelian monoid, with $[A] + [B] := [A+B]$, wh …

What is the correct terminology for the following property of a simplicial set $X_\bullet$: For every $k\geq 0$, every map $\partial\Delta^k\to X_\bullet$ can be extended to a …

Let $X$ be an abstract simplicial complex. Recall that the usual simplicial chain complex for $X$ is defined as follows. Let $C_k(X)$ be the quotient of the free abelian group on …

Let $S$ denote the category of projective (left) $R$-modules with isomorphisms for arrows. We have that $BS^{-1}S \sim B \text{GL}(R)^+ \times K_0(R)$ In proving this, in Sri …

Does anyone know a reference or proof for the following? Let $k\geq 1$ and let $X$ be a space. There is a $k$-fold multisimplicial set whose simplices in degree $n_1,\ldots,n_k$ …

If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equ …

I am trying to understand the relationship between the simplicial path space and loop space with the path space of a topological space, and the loop space of a topological space. …