Questions tagged [simplicial-stuff]

For questions about simplicial sets (functors from simplex category to sets), simplicial (co)algebras and simplicial objects in other categories; geometric realization, model structures, Dold-Kan correspondence etc. Please do not use for questions about geometry of simplices nor about triangulations.

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On proving that the geometric realization of a simplicial set is a CW complex

As the title states, I am trying to prove that the geometric realization of a simplicial set $X : \Delta^{op} \to \mathsf{Set}$ is always a CW complex. I apologize in advance for asking more than one ...
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11 views

What's the difference between an anodyne map and a trivial monomorphism?

This sounds silly but they seem the same to me and I can't find any reference confirming or denying this. The argument in my head goes like this: In the standard model structure on simplicial sets, ...
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1answer
64 views

Does categorical equivalence commute with colimit?

Lurie defines $\mathfrak{C}:Set_\Delta \rightarrow Cat_\Delta$ the functor from simplicial sets to simplicially enriched categories. And defines: Simplicial sets $X$ are considered to be ...
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22 views

Degeneracy maps in topological simplices.

Im trying to find a reference on how to define the degeneracy maps $\Delta^{n+1} \rightarrow \Delta^n$ making the simplices into a cosimplicial topological space. The face maps $\Delta^n \rightarrow \...
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Kan extension along any functor implies adjoint exists

I trying to decide whether the existence of a, say, a left Kan extension along $i: C \to D$ for any functor $F: C \to E$ implies that $i$ has a right adjoint. I have proven the converse, and I know ...
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33 views

Homotopy category of quasicategory

Let $h:qCat \rightarrow Cat$ be the fundamental/homotopy category functor, which is the left adjoint functor to the nerve functor $N$ which sends a category to simplicial $N C_n=Hom([n],C)$. I ...
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1answer
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HTT, 5.4.1.1, Lurie.

Lemma. 5.4.1.1 Lurie states that if $C$ is a simplicial category and $f_0:\partial \Delta^n \rightarrow N(C)$ is a map. $X=f_0(\{0\}), Y= f_0(\{n\})$, there is an induced map $$g_0:\partial (\Delta^1)...
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24 views

What is the hom space in over category of simplicial sets?

Let $Set_\Delta$ denote category of simplicial sets, which is enriched in $Set_\Delta$. Let $B$ be a simplicial set. We can form the over category. $(Set_\Delta)_{/B}$. Then is this also enriched ...
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35 views

A functor between $S \rightarrow S_*$ (of quasicategories)

On p.60 of Moritz Groth short course on $\infty$-category he writes, If $C$ is an $\infty$-category adding a disjoint base point defines a functor $$_+:C \rightarrow C_{*/}$$ where $C_{*/}$ is the ...
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Reference for homotopy colimits

In the first paragraph of the following paper by Heuts and Moerdijk they speak of A well known construction of homotopy colimits $$h_{!}:sSet^A \rightarrow sSet/NA$$ $A$ is small category. $sSet$...
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What are small $(\infty,1)$ categories?

The term first appeared in Chapter 3 of Lurie's HTT, p144. Where he says The simplicial category $Cat^\Delta_\infty$ has as objects (small) $\infty$-categories. What are small $\infty$-...
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Showing that $\pi_n(X,v)$ satisfies inverse axiom.

Given a fibrant simplicial set $X$ (has lifting condition with all horns) and a vertex $v \in X_0$. I want to show that the simplicial homotopy group $\pi_n(X,v)$ satisfies the inverse axiom. ...
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Product of categorical equivalences.

Let $S,T$ be simplicial sets. Suppose $f:S\rightarrow T, f':S'\rightarrow T'$ are both categorical equivalence of simplicial sets. In the sense of Lurie, i.e. the induced map $\mathfrak{C}$ is an ...
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20 views

HTT, 2.2.5.2, Lurie

Lemma 2.2.5.2: Every inner anodyne $f:A \rightarrow B$ is a categorical equivalence. The proofs states that it is sufficient to show: $\mathfrak{C}[f]$ is a trivial cofibration of simplicial ...
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A Simplicial Proof of Eilenberg-MacLane spaces representing cohomology?

Is the following sketch of a proof of "$K(G,n)$ represents $H^n(-;G)$" correct? I feel like it should be well-known, but my search didn't find anything. Suppose $X$ is a CW complex. Singular ...
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12 views

n-morphisms between (n-1)-morphisms who don't share a boundary

I'm looking for a reference to a construction of an "$\mathbb N$-Category" whose n-morphisms are morphisms between (n-1)-morphisms that may not have the same co/domains. That is, we might have a 2 - ...
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13 views

Uniqueness of a composition of degeneracy maps

Given a simplicial set $X$ and two surjective, order-preserving maps $\alpha, \beta: [n]\to [m]$ (which are compositions of degeneracy maps) inducing maps $\alpha^*, \beta^*:X[m] \to X[n]$, and some ...
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Reference request on “Simplicial” sets with all maps instead of only monotone ones.

Let $[n]=\{0,\dots,n-1\}$, $[0]=\emptyset$ and $\Sigma_{mn}$ be a set of all maps from $[m]$ to $[n]$, ($\Sigma_{0n}$ consists of a single map and $\Sigma_{no}=\emptyset$ for all $n>0$). Also let $[...
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1answer
55 views

What is the relation of a monoid and a topological monoid

Let $X$ be a simplicial set. We say that $X$ is a $H$-space if it has a map $m:X\times X\to X$ and a point $e\in X$ which is a homotopy identity, that is, the map $m(e,-),m(-,e):X\to X$ are homotopic ...
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1answer
45 views

Do colimits/limits exist in category of enriched categories?

This question may be too general. I am interested in references or proofs for special cases. I follow the definition in Chapter I of Kelly's Enriched Category. Let $V$ be a monoidal category theory. ...
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Is the join operation on simplicial sets compatible with the join operation on categories?

The join $\mathcal{C} \star \mathcal{C'}$ of two categories $\mathcal{C}, \mathcal{C'}$ is defined here: https://ncatlab.org/nlab/show/join+of+categories A key feature is that if $X$ (resp. $Y$) are ...
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40 views

HTT, 2.4.1.3, Lurie

Lurie claims the following is a formal consequence. [2.4.1.3] Let $p:X \rightarrow S$ be an isomorphism of simplicials. Then every edge of $X$ is $p$-cartesian. Definition of $p$-Cartesian is ...
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11 views

A characterization of inner fibration by fibers.

In Lurie's HTT, pg 97 He wrote, $p:X \rightarrow N(C)$ is an inner fibration if and only if $X$ is an $\infty$-category. Then it follows readliy $p:X\rightarrow S$ is an inner fibration iff the fiber ...
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22 views

Proof that higher homotopy groups of kan complexes are abelian using Eckmann-Hilton

I try to prove that higher homotopy groups of kan complexes are abelian using an Eckmann-Hilton argument. For the definitions I followed the book "Simplicial objects in algebraic Topology" by Peter ...
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19 views

HTT, 1.2.2.3, Lurie: $Hom^R_S(x,y)$ is a Kan Complex.

p.27. Let $S$ be an $\infty$-category, let $Hom^R_S(x,y)$ denote the space of right morphisms from $x$ to $y$ whose $n$-simplices is the set of all $z:\Delta^{n+1} \rightarrow S$ such that $z|\Delta^{...
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1answer
43 views

HTT, 2.1.1.3, Lurie

On Lurie's Higher Topos Theory, Prop. 2.1.1.3, Let $F:C \rightarrow D$ be a functor between categories. Then $C$ is cofibered in groupoids over $D$ if and only if the induced map $N(F):N(C)\...
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On weak equivalences of marked simplicial sets.

I was wondering if given a weak homotopy equivalence of simplicial sets $$f: X \to Y ,$$ after applying the maximal marking $f^{\sharp}: X^{\sharp} \to Y^{\sharp}$ we obtain a weak equivalence of ...
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1answer
37 views

Adjunction on simplicial sets I

I am trying to understand the proof adjunction described in page 244-245, of Joyals Theory of Quasicategories. Background: $$i^*:S/I \rightarrow S/\partial I = S \times S$$ $S$ is category ...
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Adjunction on simplicial sets II

I am trying to understand computation of the adjunction described in page 245, of Joyals Theory of Quasicategories. This is a follow up to my other post. Background: $$i^*:S/I \rightarrow S/\...
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1answer
78 views

On a lemma regarding non-degenerate points of (geometric realizations of) simplicial sets

I'm currently trying to understand Lemma 14.2 of Peter May's Simplicial Objects in Algebraic Topology, which (paraphrasing) states that given a simplicial set $X$ and a point in the coproduct $(x,p) \...
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1answer
30 views

Characterization of maps into joins of simplices

This is on page 67, Lem 23.14 of Rezk's notes. I can intuitively see why this is true but cannot write this out neatly. By computing component wise we may attempt to construct $f$ given a set of ...
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23 views

An element in the simplicial homotopy group

I am confused with this construction in Goerss Jardine's Simplicial Homotopy Theory, pg. 26+27 Is not $v$ a $0$-simplex? So I suppose we define $v_i$ as the composition $\Delta^n \rightarrow \Delta^0 ...
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1answer
45 views

Skeletons of simplicial set

It is claimed in Rezk's notes, Prop 15.24, pg. 46, that we have an obvious map of push out What I don't understand is the explicit top horizontal map: what is it? It seems that he is using the ...
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1answer
33 views

On proving that the geometric realization of $\Delta^n = \Delta(-,n)$ is homeomorphic to $|\Delta^n|$.

I'm trying to prove that the geometric realization of $\Delta^n = \Delta(-,n) : \Delta^{op} \to \mathsf{Set}$ coincides with the geometric realization of $\Delta^n$ as a simplicial complex (I will ...
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2answers
23 views

What are the vertices of an abstract simplicial complex?

So a simplex is the $k$-dimensional convex hull of $k+1$ vertices. The convex hulls of a subset of it's vertices are it's faces. A simplicial complex is a collection of simplices such that each face ...
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1answer
39 views

coequalizer of simplicial sets

This is the statement whose first line of proof I am confused. This is on page 10, of Goerss, Jardine's Simplicial Homotopy Theory. (i) How does one prove the "presentation" of $\partial \Delta^n$? ...
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Cech cohomology and the simplicial cohomology of the nerve of an open cover.

I am familiar with the usual nerve of a small category. Is there any way to construct the nerve of an open covering $\{U_i\}$ as the nerve of some small category? How does the homology/cohomology of ...
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1answer
20 views

Products in the homotopy category of a model category

Consider the category $\operatorname{Set_\Delta}$ of simplicial sets, equipped with a model structure such that it becomes a cartesian closed model category (e.g. the Quillen or Joyal model structure)....
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1answer
46 views

How to identify commutative diagrams by exponential law

In the category of simplicial set $S$, we have a bijection $ev_*:\hom_S(K,\operatorname{Hom}(X,Y))\to \hom_S(X\times K , Y)$ I wonder how to identify commutative diagrams using this bejection. ...
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1answer
18 views

Cofibrations induce fibrations between simplicial mapping spaces in a simplicial model category

A simplicial model category satisfies the axiom SM7: if for any cofibration $i:A\to X$ and any fibration $p:E\to B$ the map of simplicial sets (induced by the functoriality of map) $map_M(X,E)\to ...
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1answer
67 views

Derived functor of the realisation in a simplicial model category

Given a model category $C$, a simplicial symmetric monoidal model category $D$ (in the sense of Goerss-Jardine) and a left Quillen functor $F:C\to D$, define $|F|:C\to\mathsf{sSet}$ to be the functor $...
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1answer
132 views

Higher categories for category theorists?

I'm not actually a category theorist, but assume I have some background in category theory and am pretty comfortable with it; and I want to learn higher category theory (say in the sense of Boardmann-...
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1answer
29 views

Question on an example of simplicial group

Let $G \in Ab$ denote an abelian group and $X: \Delta^{op} \to Ab$ the functor such that $X([n]):=G^{n+1}$, the face maps $\delta_{n+1,i}:[n] \to [n+1]$ correspond to the projections $\pi_{n+1,i}:G^{n+...
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35 views

Why if $n\geq1$ and $0\leq i\leq j\leq n$ then $F_n^j\circ F_{n-1}^i=F_n^i\circ F_{n-1}^{j-1}$?

Why if $n\geq1$ and $0\leq i\leq j\leq n$ then $F_n^j\circ F_{n-1}^i=F_n^i\circ F_{n-1}^{j-1}$? We know that an n-simplex $\mathbb{\Delta}_n$ is a subspace of $\mathbb{R}^{n+1}$ given by $\mathbb{\...
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1answer
47 views

Horn Filling Condition in Quasicategories

A simplicial set $X$ is called a Kan complex if every horn $\Lambda^k [n]$, $0 \leq k \leq n$ has a filling: Here $\Lambda^k [n]$ is the union of the faces $\Delta[n]_i$ with $1 \leq i \leq n$, $i \...
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3answers
59 views

I cant simplify this A’B’C + A’BC + A’BC’ + AB’C + ABC boolean expression to A'B+C

I have to get this expression A’B’C + A’BC + A’BC’ + AB’C + ABC to A'B+C. I did this but i cant finish itm i dont know how to. A’B’C + A’BC + A’BC’ + AB’C + ABC A'B(C+C')+C(A'B'+AB'+AB) A'B+C(A'B'+...
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1answer
39 views

Weak $n$-Category

I'm trying to obtain an intuition for weak n-categories based on explanations from: https://en.wikipedia.org/wiki/Higher_category_theory#Weak_higher_categories https://ncatlab.org/nlab/show/n-...
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1answer
73 views

Proof all anodyne extensions come from saturating $\Lambda_I^0({}, M)$

In Prop 3.1.2 in Cisinski's Higher Categories and Homotopical Algebra, he claims $l(r(\Lambda_I(\{\}, M)) = l(r(\Lambda_I^0(\{\}, M))$, where $M$ is the cellular model and (defined in 2.4.14), $\...
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0answers
45 views

Why does Spivak define the realization functor from fuzzy simplicial sets to extended pseudo metric spaces the way he does?

In Spivak's paper on metric realization of fuzzy simplicial sets, he sends a fuzzy $n$-simplex of strength $a$ to the set $$ \{(x_0,x_1,\dots,x_n) \in \mathbb{R^{n+1}} |x_0+x_1+\dots+x_n = -\lg(a) \} $...
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1answer
41 views

Equivalence between the Dwyer-Kan loop groupoid and the fundamental groupoid

Let $X$ be a homotopy 1-type (a space with vanishing homotopy groups above degree one). It is a classical fact that $X$ can be recovered completely from its fundamental groupoid. On the the other ...