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-Kahn correspondence etc. Please do not use for questions about geometry of simplices nor about ...

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2
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1answer
33 views

explicit equivalent relation in the expression of the classifying space of a monoid

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. (It's also called the internal nerve.) The ...
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1answer
45 views

what means 'the realization of a topological category'

In the paper Homology Fibrations and the "Group-Completion" Theorem. page 280 bottom line 10-bottom line 12, what means 'the realization of a topological category'?
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vote
1answer
23 views

homotopy module of a simplicial module

I'm reading a paper about the cotangent complex and I'm having trouble with one of the definitions (3.4 of http://homepages.math.uic.edu/~bshipley/iyengar.pdf ). Let $V$ be a simplicial $R$-module. ...
6
votes
1answer
99 views

cofaces and codegeneracies on Simplicial Sets

Let Δ be the category of finite ordinal numbers with order-preserving maps, i.e., Δ consists of objects strings A morphism f:n→[m] is an order-preserving function (a functor) and we can think of the ...
2
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1answer
83 views

Induced map on simplices from order-preserving maps between finite ordinal numbers

Let $\Delta$ be the category of finite ordinal numbers with order-preserving maps, i.e., $\Delta$ consists of objects strings $[n]: 0→1→2→⋯→n$. A morphism $f:[n]→[m]$ is an order-preserving function ...
3
votes
1answer
64 views

Universality of the Simplex Category. Proving Functoriality of the Map.

Let $B$ be a strict monoidal category, and $\left \langle c,\mu ',\eta ' \right \rangle$ a monoid in $B$. Now suppose we consider the simplex category $\left \langle \triangle ,+,0 \right \rangle$, ...
0
votes
1answer
90 views

Category of ordinal numbers

Let $\Delta$ be the category of finite ordinal numbers with order-preserving maps, i.e., $\Delta$ consists of objects strings $$ [n]: 0 \to 1 \to 2 \to \dots \to n. $$ A morphism $f:[n] \to [m]$ is ...
7
votes
1answer
85 views

Boundary of boundary of singular cube is zero (Spivak)

At the bottom of page 99 of M. Spivak's Calculus on Manifolds he arrives at the formula $$\partial (\partial c)=\sum_{i=1}^n \sum_{\alpha=0,1} \sum_{j=1}^{n-1} \sum_{\beta=0,1} ...
2
votes
0answers
36 views

Representation of functions in the simplex category

I am using the following characterization of the simplex category $\Delta$: The objects are finite ordinals and the arrows are weakly monotone functions $f:n\rightarrow m$. $0$ is initial and $1$ is ...
0
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0answers
38 views

Where can I find Kan's paper “On c.s.s. categories” from 1957.

Does anyone know how I can find the following article by Daniel Kan: Kan, D. M. On c.s.s. categories Bull. Soc. Math. Mexicana (1957), 82-94. Quillen lists it as a reference in his paper Rational ...
0
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1answer
37 views

Is the skeleton-coskeleton adjunction $sSet$-enriched?

Let $n\geq 0$ be an integer. Is the adjunction $$ \mathbf{sk}_k\colon sSet \leftrightarrows sSet\colon\mathbf{cosk}_k $$ of the skeleton and coskeleton an $sSet$-enriched adjunction?
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0answers
48 views

Long exact sequence of homotopy groups $\pi_n$ for a pointed homotopy pullback square

Let \begin{align} A &\to B\\ \downarrow &~~~~~\downarrow\\ C &\to D \end{align} be a homotopy pullback square of pointed simplicial sets. One gets a long exact sequence $$ ...
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0answers
55 views

Classifying space infinite totally ordered set contractible

Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This should be the (possible infinite) simplex with vertices $X$, hence I ...
2
votes
0answers
49 views

Question on the coskeleton functor $\mathbf{coskel}'$ for pointed simplicial sets

There is an abstract construction of the coskeleton functor for simplicial set as follows: Fix an integer $n\geq 0$ and take the inclusion of the full subcategory $i_n\colon\Delta_{\leq ...
3
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1answer
76 views

Does the functor $\mathbf{cosk_n}:sSet\to sSet$ preserve Kan complexes?

On this nlab page a functor $\mathbf{cosk_n}:sSet\to sSet$ is constructed. Is $\mathbf{cosk_n}(K)$ of a Kan complex $K$ again a Kan complex?
2
votes
1answer
93 views

Homology of a simplicial set

Let $X$ be a simplicial set. Define the complex $(C^X_\bullet,D)$ by $$C^X_n=\bigoplus_{X_n} \mathbb{Z}$$ and $$D_n=\sum_{i=0}^n (-1)^i d_i:C_n \to C_{n-1}$$ where the $d_i$'s are the face maps. I ...
2
votes
1answer
60 views

Why is the internal hom of a Kan complex also a Kan complex?

Let $X, Y : \Delta^{op} \to \text{Set}$ be simplicial sets. I've seen it stated that if $Y$ is a Kan complex, then the internal hom $Y^X$ is a Kan complex also. How does one show this?
5
votes
1answer
183 views

What does it mean for a category to be “tensored over” another category?

What does it mean for a category to be "tensored over" another category? I was reading "Stable model categories are categories of modules" by Schwede and Shipley ...
6
votes
2answers
193 views

How to intrinsically think about simplicial objects.

It seems that a simplicial set should be thought as a space/thing, with encoded building information. The set of $n$-simplicies is the set of n-dimensional pieces and the boundary and degeneracy maps ...
1
vote
0answers
32 views

Proof of Quillen's lemma about minimal fibrations

In Hovey's book $\textit{Model categories}$ (available online here http://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey--model-cats.pdf) one finds the classical result that any ...
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vote
0answers
66 views

Topological insight on an algebraic theory.

I recently read about the marvelous bar construction. As far as I understand, it is something of a free resolution in an abstract setting. I'm wondering if it can be used to get topological insights ...
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0answers
25 views

Simplicial Complexes of Graphs notation question

I'm studying Jakob Jonsson's book Simplicial Complexes of Graphs very rigorously and in depth. I've been okay so far with the intensity and notation, but on page Chapter 3, section 2, page 30, I'm ...
0
votes
1answer
13 views

proof of the decomposition of a fibration by a trivial fibration followed by a minimal fibration

In Hovey's book theorem 3.5.9 asserts that any fibration $p:X\rightarrow Y$ can be factored as a trivial fibration followed by a minimal fibration. The proofs begins by defining a set T of simplices ...
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0answers
34 views

Criterion for projectively cofibrant diagrams?

Let $\mathbf{D}$ be a small category, $\mathbf{Set}$ the category of sets and $\mathbf{sSet}$ the category of simplicial sets with its usual model structure. The category of functors $\mathbf{D}\to ...
3
votes
0answers
68 views

Kan's loop group construction

I'm looking for a good place to read about the loop group construction $G : \mathbf{sSet_0} \to \mathbf{sGrp}$ taking a reduced simplicial set $X$ and producing a simplicial group $GX$. I would also ...
2
votes
1answer
47 views

$n$-skeleton and the category of finite simplicial complexes

Let $C$ be the category of finite simplicial complexes with simplicial maps. Then I want to define a functor $F : C \rightarrow C_n$, where $C_n$ is the subcategory of $C$ consisting of complexes up ...
0
votes
0answers
20 views

Are levelwise endofunctors of simplicial sets homotopical?

Let $Set$ be the category of sets and $sSet$ the category of simplicial sets (the category of functors from $\mathbb{\Delta}^{op}$ to $Set$). Every functor $F:Set \to Set$ induces a functor ...
3
votes
0answers
120 views

Associativity of the tensor product of dendroidal sets

For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$. ...
1
vote
1answer
45 views

Can one drop the retracts in the definition of anodyne extensions?

Definition: Anodyne extensions(i.e. acyclic cofibration) of simplicial sets are the closure of horn inclusions under transfinite composition, pushouts, and retracts. The composition and pushouts can ...
1
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0answers
52 views

Mapping $X$ to $\text{sk}_2(X)$ while fixing $\text{sk}_1(X)$

UPDATE: The original question can be simplified to this: Given a finite simplicial complex $X$, can I find a continuous $f : X \rightarrow \text{skel}^2(X)$ that fixes $\text{skel}^1(X)$? Basically, ...
0
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0answers
25 views

homotopy of simplicial maps between infinite complexes

I have such a problem. Let $K,L$ be simplicial complexes, where $K$ is connected, countable and locally finite. In particular we can express $$K=\bigcup_{i=0}^\infty K_i,$$ where $K_0$ is some full ...
1
vote
1answer
43 views

Help identifying two connecting homomorphisms

I'm running into a little trouble with the proof of Proposition 11.1 on page 64 of the first chapter of Goerss and Jardine's book Simplicial Homotopy Theory. Proposition 11.1. Suppose $X$ is a Kan ...
0
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0answers
31 views

Restricting a continuous function between simplicial complexes to $2$-skeleton

I have a continuous function $f : X \rightarrow X$, where $X$ is an $8$-dimensional simplicial complex. I want to homotope $f|_{\text{skel}^2(X)} : \text{skel}^2(X) \rightarrow X$ to a function $g : ...
3
votes
1answer
66 views

Proposition 4.2 in Goerss & Jardine's *Simplicial homotopy theory*

I'm having trouble filling in a detail in Goerss and Jardine's book Simplicial homotopy theory. Their Proposition 4.2 claims that two classes of monomorphisms $\mathbf{B}_2\subset\mathbf{B}_3$ in the ...
1
vote
1answer
71 views

cohomology ring of a quotient space

what is the cohomology ring $$ H^*((S^3\times S^3\setminus \{e\})/(a,b)\sim (ab,b^{-1});\mathbb{Z}_2)? $$ Here the unit $e$ and the product $ab$ is of the Lie group $S^3=Sp(1)$.
1
vote
1answer
103 views

Is the Image of a Simplicial Complex under a Simplicial Map again a Simplicial Complex?

I think this is trivially true, but just wanted confirmation. A simple yes or no would be great! Thanks.
1
vote
1answer
47 views

Discrete simplicial spaces are fibrant

As the title suggests, I would like to understand why should a discrete simplicial space be fibrant. Let me be more precise. Consider the category $\textbf{sSet}^{\Delta^{op}}$ of simplicial spaces, ...
4
votes
0answers
75 views

(Co)homology of propositional logic

Sorry if this is a rather vague question, but it seemed like something that might be interesting. Let $P$ be a family of propositions, and let $\mathcal L(P)$ be the set of all compound propositions ...
7
votes
0answers
108 views

History of the term “anodyne” in homotopy theory

There is a notion of an anodyne morphism (usually of simplicial sets). This type of morphisms is used, for instance, to establish basic properties of quasicategories. But the name is quite mysterious. ...
2
votes
1answer
72 views

Contradictory Orientations of Faces in Simplicial Complexes

From what I've read, an orientation of a simplex is an equivalence class of total orders on its vertices, where equivalence means up to even permutation. An orientation on an $n$-simplex induces an ...
2
votes
1answer
48 views

What is a homotopy between bisimplicial maps

I am looking for the naive notion of homotopy. For maps $f, g: A\to B$ of simplicial sets $\Delta^{op}\to Sets$, a homotopy $H$ is a simplicial map $H:A\times\Delta^{1}\to B$ such that ...
5
votes
2answers
137 views

Model structure on sSet

Which is the model structure on $ \text{sSet} $ (category of simplicial sets) that makes $\text{sSet}$ Quillen equivalent to the category $ \text{Cat} $ (of small categories) by the adjunction ...
6
votes
0answers
170 views

Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
0
votes
1answer
30 views

Relating maximal elements of downsets to minimal elements of the complement

Denote by $\mathcal{P}(S)$ the set of non-empty subsets of a finite $S$. Suppose that $A\subset \mathcal{P}(S)$ is a downset, i.e., every subset $Q$ of any $P\in A$ is also contained in $A$. We can ...
11
votes
3answers
574 views

What is combinatorial homotopy theory?

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I've heard people ...
7
votes
0answers
127 views

Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
1
vote
1answer
130 views

Examples of the Geometric Realization of a Semi-Simplicial Complex

I am reading The Geometric Realization of a Semi-Simplicial Complex by J. Milnor and here are the definitions: I find it difficult to visualize without specific examples. Can anyone help to provide ...
1
vote
1answer
115 views

homeomorphism between a boundary and a sphere

I started another question related to this. Consider $|\Delta^n|:=\{(x_0,..,x_n)\in\mathbb{R}^{n+1}:\sum_{i=0}^n x_i=1, x_i\in[0,1]\;\forall i\}$ the geometric realization of the standard n-simplex ...
2
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0answers
33 views

The identity map $\Delta^n \rightarrow \Delta^n$ is a basis for $H_n(\Delta^n, \delta\Delta^n; R)$

Show that the identity map $\Delta^n \rightarrow \Delta^n$ is a basis for $H_n(\Delta^n, \delta\Delta^n; R)$. Here $\Delta^n$ is the n-simplex, and I know $\delta \Delta^n$ denotes its ...
2
votes
1answer
62 views

the 0-th homology of a simplical complex

Let $K$ be an (abstract) simplicial complex. The claim is: $H_0(K;\mathbb{Z})$ is always nonzero. Is this possible to prove it without any "special techniques to computing homology-groups"? ...