# Questions tagged [simplicial-stuff]

For questions about simplicial sets, simplicial (co)algebras and simplicial objects in other categories; geometric realization, Dold-Kan correspondence, simplicial resolutions etc.

750 questions
Filter by
Sorted by
Tagged with
172 views

### Higher categories using just simplicial sets

Is there a definition of $(\infty, n)$-category using just simplicial sets? This is the case for $n \leq 2$. Is the forgetful functor from saturated $n$-trivial complicial sets to simplicial sets an ...
148 views

### A fiber-like method to show equivalence of infinity categories

Suppose I have a functor of quasi-categories $f: \mathcal{C} \to \mathcal{D}$. I want to show a criterion like: "$f$ is an equivalence of $\infty$-categories if the homotopy fiber of $f$ ...
219 views

### Can one bypass the geometric realization in the definition of algebraic $K$-theory?

I believe there is no good notion of homotopy groups for an arbitrary simplicial set $S$. However, when $S$ is fibrant - meaning that $S\to *$ is a fibration - there is a definition. The singular ...
528 views

138 views

93 views

### Convolution algebra of a simplicial set

Consider a simplicial set $X^\bullet$ with face maps $d_i$ (assume the set is finite in each degree so there are no measure issues). Then given two functions $f,g:X^1\to \mathbb{C}$ one can form their ...
216 views

### Viewing simplicially the Stone space of types of a first-order theory

Let $T$ be a first-order theory, let $M$ be a monster model of $T$. For a set $B\subset M$, let $S_n(B) := M^n/\operatorname{Aut}^T(M)$ be the Stone space of complete $n$-types of $T$ with parameters ...
116 views

### Multiplicativity of $\operatorname{Tot}_n$

Suppose $R^\bullet$ is a cosimplicial ring. I think this induces a ring structure on $\operatorname{Tot}(R^\bullet)$-- I couldn't find a classical reference, but I did find a recent paper of Batanin ...
387 views

### Is the singular simplicial complex functor $\operatorname{Sing}_\bullet:\operatorname{Top} \to \operatorname{sSets}$ fully faithful for nice spaces?

$\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\Top}{Top} \DeclareMathOperator\sSets{sSets}$Is there a category of “sufficiently nice” topological spaces such ...
98 views

### Simplicial set from all orderings of simplicial complex

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 $n$...
114 views

333 views

### Higher Algebra, Section 2.2.2

I am reading Section 2.2.2 of Higher Algebra by Jacob Lurie. There are proofs which I cannot understand, so I need someone's help. First, I cannot understand the proof of Lemma 2.2.7. In the proof, ...
254 views

### HTT, Remark 4.2.4.5

In his book Higher Topos Theory, Lurie proves that in some favorable cases, functor categories of $\infty$-categories admits a rigid model. More precisely, he proves in Proposition 4.2.4.4 the ...
148 views

### Defining the classifying space of a group acting on a set

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gal{Gal}$Recall that $n$-simplices of the classifying space (simplicial set) $BG$ of a group are orbits of the diagonal action by shifts on $n+1$-...
101 views

### Does integration induce a Kan fibration between the mapping spaces of CDGAs and cochain complexes over $\mathbb{Q}$?

For two CDGAs $A$ and $B$ over $\mathbb{Q}$, the mapping space $\text{Map}_{CDGA}(A,B)$ is the simplicial set with $n$-simplices $$\Phi : A \to B \otimes \Omega^*(\Delta^n)$$ and simplices maps ...
1 vote
178 views

### The simplicial set with a unique non-degenerate simplex in each dimension

There is a unique simplicial set with a unique non-degenerate simplex in each dimension, (updated) and such that all faces of the non-degenerate simplex are non-degenerate. Does it have a name, and ...
121 views

### Do there exist smaller simplicial models of barycentric subdivisions?

Let $S$ be a simplicial complex and let $Bary(S)$ denote its barycentric subdivision. Of course, the geometric realizations of $S$ and $Bary(S)$ are homeomorphic. However, one issue that arises in ...
1k views

### Applications of the Dold-Kan correspondence

The Dold-Kan correspondence says essentially that simplicial abelian groups and nonnegative chain complexes of abelian groups are equivalent objects. While this is a very natural statement, I am not ...
99 views

### Interplay beween simplicial and Weyl algebra identities

Recall that the (first) Weyl algebra is the algebra generated by $x,y$ with the relation $xy-yx=1$. It can be realized as the algebra of differential operators on $k[x]$, where one generator acts as ...
313 views

### Why, if the geometric realisation of a simplicial map $p$ is a (topological) covering map, must $p$ be a (simplicial) covering map?

I essentially am asking for an explanation of the comment under this post by Tom Goodwillie. In the "Kerodon", Lurie defines a simplicial covering map as follows: A map $p:E\to X$ of ...
186 views

### Double complex of simplicial resolution

In his lectures on condensed mathematics on page 30 Peter Scholze speaks of the double complex of a simplicial resolution. How is this defined? In the next line, he writes that if $A_\bullet$ is a ...
215 views

### Higher Algebra, Propositions 2.3.4.5 and 2.3.4.9

I am reading the proof of Propositions 2.3.4.5 of Higher Algebra by Jacob Lurie. There is a part that I don't understand, and I need someone's help. In the book, Lurie introduces the notion of ...
632 views

495 views

### Higher Algebra, Theorem 2.4.3.18 and Remark 2.4.3.6

In his book Higher Algebra, Lurie introduces the notion of generalized $\infty$-operads ($\S$2.3.2). Roughly speaking, a generalized $\infty$-operad is a "family" of $\infty$-operads ...
138 views

### Does the forgetful functor from an over-$(\infty,1)$-category create weakly contractible limits?

Let's say that a limit diagram $\bar p:K^\triangleleft\to\def\sC{\mathscr{C}}\sC$ is a weakly contractible limit if the simplicial set $K$ is weakly contractible (in that $K\to*$ is a weak homotopy ...
296 views

### Do the various notions of morphism spaces of simplicial sets agree on the underived level?

$\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}$There are (at least) seven kinds of morphism spaces for a simplicial set $X$: The left-pinched morphism space $\Hom^L_X(x,y)$, The right-...
1 vote
119 views

### Simplicial sets and oriented simplicial complexes

$\DeclareMathOperator\Sing{Sing}$I'm writing a paper about simplicial sets and how they may “replace” simplicial complexes in some known results. To do this, we need to check that they induces the ...