# Questions tagged [simplicial-categories]

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39
questions

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### Is simplicial localization part of a Quillen equivalence between relative categories and simplicial categories?

There are many models for $\infty$-categories. One of them is relative categories – AKA categories with weak equivalences – which have a model structure due to Barwick–Kan. Another one is simplicial ...

**3**

votes

**1**answer

107 views

### Rigidification of marked simplicial sets

It is well known that there exists a Quillen equivalence,
$$\mathfrak{C}: Set_{\Delta} \rightleftarrows Cat_{\Delta}: N, \enspace \mathfrak{C} \dashv N$$
between Joyal's model structure on ...

**4**

votes

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174 views

### The k-ification of the compact-open topology for weak Hausdorff compactly generated spaces

Let CGWH be the category of weak Hausdorff compactly generated spaces; see e.g.
N.P. Strickland. THE CATEGORY OF CGWH SPACES: Preprint available from
https://neil-strickland.staff.shef.ac.uk/courses/...

**3**

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48 views

### A variation of the hammock localization

Let $W \subseteq \mathcal{C}$ be a wide subcategory of a category $\mathcal{C}$. The saturation $\overline{W}$ of $W$ is the class of maps of $\mathcal{C}$ which become isomorphisms in $\mathcal{C}[W^{...

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96 views

### Existence of tensor product of infinity operads

I am trying to show, or find a reference, for the following fact:
"Given O,P two infinity operads [in the sense of Lurie, HA, Definition 2.1.1.10], there always exist a tensor product".
In other ...

**2**

votes

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87 views

### Homotopy colimits of simplicial objects

Given a simplicial combinatorial model category $\mathcal{M}$ and a simplicial diagram $F\colon \Delta^{\mathrm{op}} \rightarrow \mathcal{M}$, is there a nice (i.e. explicitely computable) way of ...

**2**

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47 views

### Simplicial models for mapping spaces of filtered maps

Let $S$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. Hence $SIMP(S,K)$ is Kan.
Suppose that ${...

**3**

votes

**1**answer

71 views

### Simplicial models for fibrations between mapping spaces

Let $S,T$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. We let $|S|$ denote the geometric ...

**4**

votes

**2**answers

207 views

### Proof that a quasicategory is equivalent to the homotopy coherent nerve of a simplicial category

On the abstract of a paper by Emily Riehl and Dominic Verity, it is stated that
Every quasicategory arises as a the homotopy coherent nerve of a simplicial category up to equivalence.
Where can ...

**5**

votes

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86 views

### When is a bisimplicial set diagonal fibrant

Let $sSet^2$ be the category of bisimplicial sets.
In the diagonal model structure on $sSet^2$ weak equivalences are diagonal weak equivalence (i.e.$ X \rightarrow Y$ is a weak equivalence if $dX \...

**6**

votes

**1**answer

216 views

### Understanding two proofs in Dwyer and Kan article “Simplicial Localizations”

I can't understand the proofs of propositions 2.6 and 4.2 in https://www3.nd.edu/~wgd/Dvi/SimplicialLocalizations.pdf
We have a category $C$ and a family of maps $W$, and we define the ...

**5**

votes

**1**answer

219 views

### How are simplicial sets with Quillen model structure a simplicial model category?

I got very lost in checking that simplicial sets with Quillen model structure are indeed a simplicial model category.
Recall that a model category $\mathcal{M}$ is simplicial if it is enriched in $\...

**10**

votes

**1**answer

239 views

### Criterion for homotopy pullback square of simplicial categories

Assume given a pullback square of simplicial categories
$$\begin{array}[c]{ccc}
A&{\rightarrow}&B\\
\downarrow&&\downarrow\\
C&{\rightarrow}&D.
\end{array}$$
Suppose further ...

**4**

votes

**2**answers

517 views

### Filtered colimit of fibrations

In a model category $\mathcal{C}$, is the filtered colimit of fibrations, resp. trivial fibrations, a fibration, resp. trivial fibration?
Thm. 1.2.3.5 in Toen-Vezzosi's "Homotopical algebraic ...

**2**

votes

**1**answer

245 views

### Coproducts of weak equivalences

In a model category $\mathcal{C}$ admitting a forgetful functor to simplicial sets, is the coproduct of weak equivalences a weak equivalence?
Say even just coproducts indexed by $\mathbf{N}$.
A ...

**4**

votes

**1**answer

152 views

### Kan complexes and semigroups

Given a simplicial commutative semigroup:
(1) is it true that its underlying simplicial set is a Kan complex if and only if the simplicial semigroup was a simplicial group?
(2) is the constant ...

**5**

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49 views

### Is composition in a simplicially enriched category always determined by a compatible simplicial tensoring (if such exists)?

Let $C$ be a simplicially enriched category, i.e., there are
a collection of objects $ob C$,
a simplicial set $map_C(X,Y)$ for $X,Y \in ob C$,
composition maps $map_C(Y,Z) \times map_C(X,Y) \to map (...

**8**

votes

**0**answers

152 views

### Unaugmentable cosimplicial simplicial sheaves and realization functor

I'm studying the construction of the $\mathrm{Sing}$ functor in Morel-Voevodsky ``$\mathbb{A}^1$-homotopy theory of schemes'' and I was trying to understand the properties of its left adjoint, the ...

**2**

votes

**0**answers

97 views

### Does twisted arrows commute with the simplicial nerve construction?

Let $\mathcal{C}$ be a simplicial category, and let $N(\mathcal{C})$ be its simplicial nerve. We can form the category of twisted arrows as a simplicial category $TwArr(\mathcal{C})$
Now Lurie's ...

**2**

votes

**1**answer

242 views

### Does $\mathbf{Top}$ admit a simplicial structure

A simplicial category $\mathcal{C}$ is an $\mathbf{sSet}$-enriched category which is bitensored, in that the functors $\mathbf{Map}(-,X)$ and $\mathbf{Map}(X,-)$ admit left adjoints for every $X\in \...

**4**

votes

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63 views

### In which sense is the relativization functor “preferred”?

In A characterization of simplicial localization functors and a discussion of DK equivalences Barwick and Kan state that, while there is no preferred localization functor from relative categories to ...

**3**

votes

**2**answers

228 views

### Is the projective model structure simplicial?

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 ...

**2**

votes

**1**answer

217 views

### 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, $\infty$-...

**12**

votes

**2**answers

535 views

### Simple question: different definitions of Bousfield localization

I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them.
First definition: Let $\mathbf{...

**2**

votes

**1**answer

159 views

### Join of simplicial categories

Let $\mathcal{C},\mathcal{D}$ be simplicial categories.
Of course, we have the "naïve" join $\mathcal{C} \star \mathcal{D}$, which has
$$
\mathrm{Ob}(\mathcal{C} \star \mathcal{D}) := \mathrm{Ob}(\...

**3**

votes

**1**answer

158 views

### 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 ...

**8**

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234 views

### A completeness criterion for $\infty$-categories

We all know that for ordinary categories $\mathscr{C}, \mathscr{D}$ (with $\mathscr{C}$ small) the limit of a functor $F:\mathscr{C} \to \mathscr{D}$, if it exists, can be constructed by using ...

**4**

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424 views

### How to compute (co)limits of enriched categories?

I've asked this question on math.stackexchange some time ago (https://math.stackexchange.com/questions/1380176/how-to-compute-colimits-of-enriched-categories) and I received no complete answers, so I'...

**5**

votes

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353 views

### Stabilization of a generic pointed model category

Let $\mathcal C$ be a pointed model category. It is well-known that its homotopy category $\mathrm{Ho}(\mathcal C)$ is naturally a $\mathrm{Ho}(\underline{\mathrm{sSet}}_*)$-category, where $\mathrm{...

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155 views

### Internal Hom on simplicial presheaves and the preservation of cofibrant objects

1)Let $\mathcal{C}$ be a cartesian closed small category. Let $\operatorname{Map}\: : \: sPsh(\mathcal{C})\times sPsh(\mathcal{C})\to sPsh(\mathcal{C})$ be the internal Hom of simplicial presheaves, i....

**3**

votes

**1**answer

109 views

### Segal maps for Segal precategories

A Segal precategory is just a simplicial space $X:\Delta^{op} \to sSet$ such that its $0$-th space is discrete (i.e. constant). A Segal category is defined everywhere in the literature as a Segal ...

**1**

vote

**1**answer

124 views

### Homotopy invariance of Kan nerve of simplicial categories

The following question concerns the well-known paper of Dwyer and Kan "Localization of Simplicial Categories". They define a nerve for simplicial categories (with fixed set of objects $O$), by the ...

**0**

votes

**1**answer

226 views

### equivalence in simplicial category

Let $(\mathcal{C},W)$ be a category with weak equivalences. One can build from $(\mathcal{C},W)$ its hammock localization $L^{H}(\mathcal{C},W)$ which is a simplicial category $\textit{ie}$ a category ...

**1**

vote

**2**answers

286 views

### Defining degeneracies for semi-simplicial sets with inner Kan conditions

Suppose we are given a category enriched over semi-simplicial sets, i.e. for the simplices in this category we have well-defined boundary maps, but no degeneracy maps. Suppose also that in this ...

**1**

vote

**1**answer

232 views

### Homotopy limit of a cosimplicial category

Consider the usual model structure on Cat (category of small categories).
Which are the fibrations of the injective model structure on the category of cosimplicial categories $Fun( \Delta ,Cat )$?
...

**6**

votes

**1**answer

287 views

### Joins of, and limits in, $(\infty,1)$-categories via profunctors

I'm trying to interpret the join of $(\infty,1)$-category in a more conceptual way. Let me try to explain what I have in mind.
In the classical setting it is almost a triviality to express the join ...

**5**

votes

**1**answer

249 views

### About elegant Reedy categories

I discovered today the notion of elegant Reedy category introduced in the paper Reedy categories and the $\Theta$-construction of Julia E. Bergner and Charles Rezk. An interesting property of such ...

**4**

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314 views

### Simplicial localisation and infinity categories

If $(\mathcal{C},W)$ is a category with weak equivalences then we may naturally form its Dwyer-Kan simplicial localisation $L(\mathcal{C}, W)$. This is a simplicial category which naturally gives a ...

**4**

votes

**1**answer

294 views

### K theory of a simplicial monoidal category, Cofinality theorem

Let $X=(d\mapsto X_d)$ be a simplicial symmetric monoidal category. We define the $K$-theory space of $X$ to be $K(X)=|d\mapsto K(X_d)|$, the geometric realisation of the simplicial space $d\mapsto K(...