1
vote
0answers
61 views

When can I compute the simplicial mapping space from a presheaf to a simplicial presheaf naively?

Suppose that $\mathscr{C}$ is is a small category, $Y$ is a presheaf (of sets) on $\mathscr{C},$ and $X_\bullet$ is a simplicial presheaf. There is a spectrum of simplicial model structures on ...
6
votes
0answers
110 views

When does a sheaf of categories represent a homotopy sheaf?

Suppose that $F$ is a sheaf of categories (on a Grothendieck site or even a topological space). By this, I mean a sheaf in the naive 1-categorical sense, so it can equivalently be viewed as a category ...
6
votes
1answer
212 views

Are there models for homotopy colimits and limits of simplicial sets that generalize Kan's suspension and loop functors?

Consider the category C of pointed simplicial sets. The pair of functors X∈C↦X∧S¹∈C and Y∈C↦Map(S¹,Y)∈C models the suspension and loop functors on the underlying ∞-category of C. There is another ...
9
votes
0answers
139 views

Homotopy theory of suplattices

In Quillen's monograph Homotopical algebra where he introduced the notion of model category, he showed that if $C$ is a bicomplete category with enough regular-projectives in which either (*) every ...
4
votes
1answer
230 views

Is there a class of simplicial sets whose weak homotopy type is preserved by symmetrization?

We have the categories, $S$, of simplicial sets and $SS$, of symmetric simplicial sets (whose simplices are unordered). There are functors: $H:S\to SS$ forgetting the ordering on simplices and ...
4
votes
0answers
166 views

Is there a local projective model structure on simplicial sheaves? What are its fibrant objects?

Consider a site S (I am mostly interested in hypercomplete sites, e.g., the site of smooth manifolds). The category of simplicial presheaves SPSh(S) on S can be equipped with the local projective ...
7
votes
3answers
429 views

When is the projective model structure cartesian? When is the internal hom invariant?

If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equivalences are the ...
5
votes
1answer
302 views

The Quillen model structure on simplicial sets as a Bousfield localization

Starting with the trivial model structure on the category of simplicial sets (that is the weak equivalences are exactly the isomorphisms and the cofibrations and fibrations are arbitrary maps), is it ...
3
votes
1answer
361 views

When is homotopy orbit space weakly equivalent to orbit space, other than situation of free action?

Let $M$ be a closed symmetric monoidal model category. Let $X$ be a cofibrant object (it can also be fibrant if you like) and let $\Sigma_n$ act on $X^{\otimes n}$ by permuting the factors (note that ...
6
votes
2answers
530 views

Connective spectra versus simplicial abelian groups - very basic question

Hello, I have very general , "introductory" questions (It is quite hard for me to seek for specific things in the algebraic topology literature). I guess that connective spectra have a model ...
3
votes
2answers
454 views

Reference request - CDGA vs. sAlg in char. 0

Hello, Are the model categories of simplicial commutative algebras over $k$ and that of commutative differential graded algebras (in negative cohmological dimension) Quillen equivalent in char. 0 (or ...
1
vote
2answers
410 views

About a statement in Jardine and Goerss “Simplicial Homotopy Theory”

Hello, This probably just technical, but anyway: In "Simplicial Homotopy Theory" by Goerss and Jardine, chap. III, par. 2, after cor. 2.12, they describe a model structure on $Ch^{+}$, the category ...
2
votes
1answer
178 views

local model structure on simplicial presheaves

Hello, Let $\mathcal{C}$ be a (small) category equipped with a Grothendieck pretopology. Let $sPSh(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, together with its ...
5
votes
0answers
199 views

Left adjoints to several inclusions of homotopy simplicial model categories

The left adjoint to the inclusion $sGrp\hookrightarrow sPtSet$ of the category of simplicial groups into the category of pointed simplicial sets is homotopy equivalent to the loop suspension functor ...
7
votes
2answers
598 views

Do simplicial objects in a Topos form a model category?

Sometimes people say "If you don't like the word 'topos', just think the category of Sets", but I'm not sure to what extent this analogy holds. The real question here is, do simplicial object in a ...
10
votes
1answer
385 views

The weak equivalences in the covariant model structure

Let $S$ be a simplicial set. Recall that there is a model structure, called the covariant model structure (see HTT ch. 2 and this question), on $\mathbf{SSet}/S$ such that: The cofibrations are the ...
9
votes
1answer
796 views

Is the simplicial completion of a localizer always a bousfield localization of the injective model structure?

Background Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following ...
2
votes
0answers
293 views

simplicial deRham complex and model category structure

To every simplicial manifold is associated its simplicial deRham complex. Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) ...
4
votes
2answers
307 views

Simplicial presheaves that are colimits of themselves?

Suppose $C$ is a small category and $X_{\bullet}$ is a simplicial object in $C$. In particular, by composing with Yoneda $$y:C \to Set^{C^{op}}$$ $y(X)_{\bullet}$ is a simplicial presheaf. I believe ...
3
votes
2answers
492 views

Inner hom and geometric realization.

I would like to prove the following fact, which I learned from a previous MO question. Let $S_\cdot,T_\cdot\in\mathbf{sSET}$ be simplicial sets, and assume that $T_\cdot$ is Kan. Then there is a ...
3
votes
2answers
336 views

Model categories of simplicial objects

If $\mathcal{C}$ is a category, then surely the category of simplicial objects $s\mathcal{C}$ is not automatically a model category. What conditions must $\mathcal{C}$ satisfy in order for ...
4
votes
1answer
230 views

Compatibility of classifying space with inner-hom?

Let $\mathbf{sTop}$ be the functor category $\mathbf{Top}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let $\mathbf{sCat}$ be the functor category $\mathbf{Cat}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let ...
5
votes
1answer
764 views

Fibered/cofibered higher categories, relative model structures, slicing, and (∞,2)-category theory

Jacob Lurie defined a model structure on the category of marked simplicial sets sliced over a fixed simplicial set $S$ called the cartesian model structure. (For a definition, see here or HTT ...
19
votes
4answers
1k views

Model structure on Simplicial Sets without using topological spaces

The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...
10
votes
1answer
315 views

Is the model category of Complete Segal Spaces right proper?

Well, the title is self-explaining, I guess - I am referring to the complete Segal space model structure of Theorem 7.2 in Rezk's article "A model for the homotopy theory of homotopy theories". Has ...
8
votes
2answers
813 views

Example of a CW complex not homeomorphic to the realization of a simplicial set?

I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't ...
3
votes
1answer
542 views

Verifying a technical lemma regarding homotopy pushouts in the theory of simplicial model categories

Important Edit: I e-mailed Jacob Lurie, and he said that the statement of condition (*) is incorrect as printed. Here is the correct statement of (*): For any cofibration $f:A\to B$ and ...
7
votes
4answers
472 views

An explicit description of Lawvere's segment in the category of simplicial sets

In any presheaf topos, there exists an object called Lawvere's segment, which can be described as the presheaf $L:A^{op}\to Set$ such that for each object $a\in A$, $L(a)=\{x\hookrightarrow\ h_a: x\in ...
14
votes
1answer
651 views

Is there a combinatorial way to factor a map of simplicial sets as a weak equivalence followed by a fibration?

Background on why I want this: I'd like to check that suspension in a simplicial model category is the same thing as suspension in the quasicategory obtained by composing Rezk's assignment of a ...
15
votes
1answer
1k views

Homotopy colimits/limits using model categories

A homotopy (limits and) colimit of a diagram $D$ topological spaces can be explicitly described as a geometric realization of simplicial replacement for $D$. However, a homotopy colimit can also be ...
8
votes
3answers
829 views

Infinity groupoid objects

I was wondering if there is a model-theoretic way of defining the infinity category of infinity-groupoid objects in a category $C$ (more generally, if $C$ is an infinity category itself, but, right ...
0
votes
1answer
270 views

Analogs of left, right, inner, and Kan fibrations in CGWH

It is a theorem that the category of compactly generated weakly Hausdorff (CGWH) spaces is Quillen equivalent to the category of simplicial sets with the Kan model structure. However, I know next to ...
4
votes
2answers
370 views

Motivation for the covariant model structure on SSet/S

I was reading HTT 2.1.4, and I just totally lost what was going on. Could someone provide some motivation for this section? Why do we want another model structure? I'm sorry for not providing ...
18
votes
4answers
1k views

A Peculiar Model Structure on Simplicial Sets?

I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do ...
9
votes
3answers
721 views

What are the fibrant objects in the injective model structure?

If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial ...
5
votes
3answers
374 views

Abstract Relation between Presehaves and Simplicial Sets

Every presheaf (let's say on a topological space) comes with restriction maps. The open sets of a topological space are ordered by inclusion and these inclusions yield the restrictions. Now a sheaf ...