The simplicial-stuff tag has no usage guidance.

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### Functorial Path objects for bisimplicial sets

Let $s^2Set$ denote the category of bisimplicial sets with the Bousfield-Kan model structure. Recall that a path object is a factorization of the diagonal $X\rightarrow X\times X$ into $X\xrightarrow{...

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### Simplicial Model Categories

Let $D$ be a combinatorial simplicial model category (e.g $SSet$ with the standard model structure) and let $C$ be a small simplicial category. Of course, we can consider the projective model ...

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### Moerdijk Model Structure on Bisimplicial sets

Let $s^2Set$ denote the category of bisimplicial sets, i.e. simplicial objects in the category of simplicial sets. Recall that in the Moerdijk model structure on $s^2Set$, weak equivalences are "point-...

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### A model structure on semi-simplicial algebraic Kan complexes?

By an algebraic semi-simplicial kan complex I mean a semi-simplicial set (i.e. a presheaf on the category of finite ordered sets and injective order preserving maps), which is a Kan complex (in the ...

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### Presheaves of Dendroidal Sets?

Are there any references available for presheaves of dendroidal sets?
Seems like a natural extension of simplicial presheaves.

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### Left adjoint to Double Nerve?

The well known nerve functor from small categories to simplicial sets has a left adjoint, namely the fundamental category functor. Does the double nerve functor $N^2:2Cat\rightarrow sSSet$ from 2-...

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### Can relative homotopy groups be represented as relative homology groups of some Moore complex?

Daniel Kan defined a combinatorial version of the homotopy group $\pi_n(X)$ of a simplicial set $X$ as the $(n-1)$st homology of the (non-commutative) Moore complex $\tilde{G}(X)$, where $G_iX$ is ...

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### Errata on Rezk's paper

I am reading this paper of Rezk's http://arxiv.org/abs/0901.3602 A cartesian presentation of weak n-categories, and as it is pointed out in the introduction, it contained a wrong statement (2.19 in ...

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### “Sheaf” on the nerve of a category

So I have the nerve of a category and want to communicate information about the sets at each level in the simplicial set (nerve). Is there a way to regard the nerve as a category itself? Then one ...

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### Are the unwound thin realization and fat realization homotopy equivalent?

This is a question about a theorem (proposition 2) in the article--On the homotopy type of classifying spaces
Recall some definitions first:
Given a category $\mathcal{C}$ internal in $\...

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### Is this a model for $K$-theory of a triangulated category?

The recent question Complete the following sequence: point, triangle, octahedron, . . . in a dg-category reminded me of something I wanted to clarify long time ago; most likely this is now well known ...

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### Complete Segal operads and dendroidal sets

There is a Quillen equivalence between the model category presenting Lurie's $\infty$-operads (which are inner fibrations $\mathcal{C}\to\mathrm{N}(\mathbf{F})$ satisfying certain conditions) and the ...

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### What do globes (used to construct globular sets, $\omega$-categories, etc.) actually look like?

Nlab introduces the globular category as a geometrical model to construct certain higher categorical structures (e. g. strict $\omega$-categories), just as quasi-categories, for example, are modelled ...

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### Two natural maps asssociated with the nerve of a cover

Let $X$ be a nice (e.g. paracompact, locally contractible) topological space, and let $\mathcal{U}=\{U_i\}_{i\in I}$ be an open cover of $X$. Also denote by $N$ the (topological realization of) the ...

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### From relative categories to marked simplicial sets

Both relative categories and marked simplicial sets (over Δ^0) present the ∞-category of ∞-categories.
Naturally, one could ask whether there is a reasonably direct way to pass between these two ...

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### Relative Hurewicz Theorem

For a given zero-reduced simply connected simplicial set $X$, one can define simplicial group $GX$ representing the loop space of $X$, its Abelianization $AX = GX/[GX,GX]$ and show that the map ...

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### pullback square in Goerss-Jardine

In proving the the existence of the Reedy model structure on the category of simplicial objects in a model category $\mathcal{C}$, Goerss-Jardine prove there is a pullback square induced by a map of ...

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### Parallel transport on simplicial manifold? [closed]

Do you know some reference about the notion of parallel transport for simplicial manifolds?

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### Homology of simplicial manifolds

Let $M_{\bullet}$ be a simplicial manifold. There are two ways to computing its cohomology. Consider the cosimplicial module $A_{DR}(M)$. It defines a functor $A_{DR}(M_{\bullet})\: : \: \boldsymbol{\...

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### The properness of the special singular simplicial spaces

This is a question related to another one in MO
Background : A special simplicial space $X_{\cdot}$ is a simplicial space with $X_{0}=\ast$ and $X_{n}\simeq X_{1}^{n}$ via the simplicial map $v_{i}:[1]...

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### Path space of a simplicial topological space?

Given a connected topological space $X$, its space of path $PX$ is again a topological space. On the other hand, for a simplicial set $K_{\bullet}$, its path space is given by
$$
PK_{n}:=\...

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### Is the hom-simplicial set in the hammock localization a nerve?

Let $(C,w)$ be a relative category. Then associated to it we have its hammock localization, $L^H(C,w)$, which is a simplicially enriched category.
If $X,Y\in C$, the description of the simplicial set ...

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### Bott-Samelson theorem for simplicial sets

Let $X\in \mathrm{sSet}$ and $FX$ be the Milnor's construction (model for $\Omega\Sigma |X|$) - in each dimension $n$ this is the free group on $X_n$ with one relation $*=1$. I'm interested in $\...

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### cohomology ring of compact submanifolds of Euclidean spaces

Suppose we have a compact $m$-dimensional submanifold $M$ of $\mathbb{R}^N$ and we want to know the cohomology ring $H^*(M;\mathbb{Z})$.
Let $\epsilon>0$ and a $m$-dimensional finite simplicial ...

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### Basic technical things about simplicial sets to have a good understanding of quasicategories

May someone provide me the list of basic techniques about simplicial sets, in order to have a good understanding of the definition of a quasicategories, $\infty$-topos, $\infty$-stacks, $\infty$-...

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### Test categories applied to Dold-Kan correspondence?

Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category $X$ is a category that can be used in place of the simplex category $\Delta$ to ...

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### Grothendieck - A group as a sheaf over simplicial complexes

In this blog post, Terence Tao gives the following definition of a group.
Definition. A group is (identifiable with) a (set-valued) sheaf on the category of simplicial complexes such that the ...

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### P.G.Goerss, J.F.Jardine, “Simplicial Homotopy Theory” prerequisites

I know that such questions may be better suited for math.stackexchange, but I believe that that the topic of simplicial homotopy theory is advanced enough for mathoverflow.
Besides, I know that there ...

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### Is this almost-cosimplicial object familiar?

I have a sequence $X_0,X_1,\ldots$ of Abelian groups, along with face maps $d_0,\ldots, d_n: X_{n-1}\to X_n$ which satisfy the standard cosimplicial identities EXCEPT at the very bottom, where in $\...

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### Categorical or simplicial introduction to modern homotopy theory

I've heard some people saying that it would be easier to study homotopy theory first through simplicial sets. First, i thought that these people were not being serious. But it caught me wondering...
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### Extending a left fibration along an inner horn

Let $\Lambda^n_i \subseteq \Delta^n$ be an inner horn, and let $X \rightarrow \Lambda^n_i$ be a left fibration. Does there exist a left fibration $Y \rightarrow \Delta^n$ such that $X = Y \times_{\...

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### Cofibrancy of simplicial objects [duplicate]

Let $\mathcal{C}$ be a site. Consider $sPsh(\mathcal{C})$ be the equipped with the local projective model structure. Let $C_{\bullet}$ be a cofibrant object in $\mathcal{C}$ and let $y(C_\bullet)$ be ...

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### Symmetric sub-simplicial sets of the nerve of $\mathbb{N}$

The nerve of $\mathbb{N}$ is the simplicial set $N\mathbb{N}$ with
simplices: tuples $(k_1,\ldots, k_r)$ with each $k_i\in \mathbb{N}$
degeneracies: inserting $0$
faces: adding consecutive entries ...

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### explicit description of the cosimplicial simplicial set $Q^{\bullet}$

I'm struggling to understand the explicit description of the cosimplicial simplicial set $Q^{\bullet}$ on page 76 (section 2.2.2) of Lurie's book Higher Topos Theory, and would be grateful if someone ...

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### What is this construction using iterated face maps of semisimplicial sets?

Let $X$ be a semisimplicial set (face maps but no degeneracy maps). Fix a positive integer $k$. Let $Y_n$ be $X_{(n+1)k}$ and then define $\partial^Y_i:Y_n\to Y_{n-1}$ by
$$\partial^Y_i = (\partial^...

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### Polynomial differential forms on $BG$

Let $\Omega^{*}_{\text{poly}}\: : \: sSet\to dg_{\geq 0}Comm_{+}$ be the polynomial De Rahm functor on simplicial sets, where the codomain is the category of commutative differential graded algebras ...

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### Item (4) in Lurie's definition of the class of marked anodyne morphisms

I have a question concerning Remark 3.1.1.3 in Lurie's "Higher Topos Theory" about the definition of the class of marked anodyne morphisms. There, it is mentioned that "it suffices to allow $K$ to ...

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### Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting)

Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its fibrewise suspension, i.e. its suspension as an object of the slice $\...

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### Basic questions about simplicial commutative rings

I am trying to learn about simplicial commutative rings, and would be grateful if one can help with some basic facts about them. Basically, I would like to understand how to do homological algebra ...

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### For a simplicial set $X$, is the category of non-degenerate simplices of $X$ a full subcategory of the category of simplices of $X$?

I'm slightly confused, but I think you can help me. Let $X$ be a simplicial set. The category of simplices of $X$ and its subcategory of non-degenerate simplices are defined at http://ncatlab.org/nlab/...

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### $E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral ...

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### Does the fat geometric realization take limits to homotopy limits?

I am deeply confused about geometric realizations and finite limits.
Suppose that I am working with simplicial sets (I dont need simplicial spaces) that are "good" in the sense of Segal so that I can ...

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### Is the dg-nerve functor a Quillen equivalence?

Lurie defines the dg-nerve $N_{dg}(\mathcal{C})$ of a dg-category $\mathcal{C}$ in Higher Algebra Construction 1.3.1.6: for each $n \geq 0$, we define $N_{dg}(\mathcal{C})_n\simeq \text{Hom}_{\mathcal{...

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### Does a topological hypercover always have free degeneracies?

This question arises when I am reading Dugger and Isaksen's "Hypercovers in topology". According to Definition 4.1 in that paper, A hypercover of a space $X$ is an augmented simplicial space $U_*\to X$...

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### Why should we regard $PL(M)$ as a simplicial group?

Let $M$ be a manifold. If $M$ is smooth, it is clear what $\text{Diff}(M)$ should be, namely it should be the set of diffeomorphisms of $M$ equipped with the topology in which a sequence of ...

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### Is the simplicial set $(\Delta_3/\partial \Delta_3)^{\Delta_1}$ finite?

A simplicial set is called finite if it has only a finite number of non degenerate simplicies (or equivalently, if its number of $n$-simplicies grows polynomially with $n$). In an answer to a ...

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### Equivalent definition of a Kan fibration

It is known (follows for example from Proposition 4.2 of Simplicial Homotopy Theory by Goerss and Jardine) that a Kan-fibration can be defined as a map having the right lifting property with respect ...

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### Natural transformations induce homotopies - Is this true in the “fat” world?

Let $\mathcal{C}, \mathcal{D}$ be categories internal to topological spaces (or compactly generated Hausdorff spaces, if you like) $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ be continuous functors ...

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### Does the fat realization of simplicial spaces commute with finite limits up to homotopy?

I'm willing to work in the category of compactly generated Hausdorff spaces.
The ordinary geometric realization of a simplicial space commutes with taking pullbacks and preserves the terminal object, ...

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### Does it require Reedy fibrancy when we want the totalization to be weakly equivalent to the homotopy limit?

This question arises when I am reading the last two Chapter of Hirschhorn's "Model categories and their localizations"
In Part (2) of Theorem 19.8.4 of that book it says
If $(\bf{\Delta},\mathcal{M})...