User philippe gaucher - MathOverflow

Cofibrant replacements of a given object in a combinatorial model category

2013-04-29T09:00:08Z

In a combinatorial model category, every $\lambda$-filtered colimit is a homotopy colimit for $\lambda$ regular big enough. So for $\lambda$ regular big enough, every $\lambda$-filtered colimit of a diagram of cofibrant replacements of a given object $X$ is a cofibrant replacement of $X$. Does the contrary hold, i.e. is the full subcategory of cofibrant replacements of a given object accessible ?

EDIT : the class of cofibrations is accessible so for $\lambda$ regular big enough, a $\lambda$-filtered colimit of cofibrations $\varnothing \to X_i$ is a cofibration ; the class of weak equivalences is accessible so for $\lambda$- regular big enough, a $\lambda$-filtered colimit of weak equivalences $X_i\to X$ is a weak equivalence ; so for $\lambda$ regular big enough, a $\lambda$-filtered colimit of cofibrant replacements of $X$ is a cofibrant replacement of $X$.

Answer by Philippe Gaucher for Cofibrant replacements of a given object in a combinatorial model category

2013-05-10T10:08:24Z

(sorry I have troubles with comments, I post here even if it is not an answer) I have a new information. In On a fat small object argument, it is proved that in a λ-combinatorial model category, every cofibrant object is a λ-filtered colimit of λ-presentable cofibrant objects, which is close to what I wanted.

Topological question about right-lifting property and the evaluation map

2013-04-25T08:14:55Z

Let $Z$ be a $\Delta$-generated space (a colimit of simplices -not sure that this hypothesis is important but it is the framework I am working in). The set of continuous maps $Z^{[0,1]}$ from $[0,1]$ to $Z$ is equipped with the $\Delta$-kelleyfication of the compact-open topology (the internal hom of the category, see below).

QUESTION : Suppose that the evaluation map at $0$ from $Z^{[0,1]}$ to $Z$ satisfies the right lifting property with respect to any monomorphism of $\Delta$-generated spaces (i.e. injections). Then I suspect that $Z$ must be discrete. Any counter-example ?

Concerning $\Delta$-generated spaces, a short bibliography, just in case that it is important : 1. Notes on Delta-generated spaces by Dugger : http://math.uoregon.edu/~ddugger/delta.html 2. The proof that they assemble to a locally presentable category : A convenient category for directed homotopy by Fajstrup-Rosicky : http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html 3. A survey of their properties : Section 2 of Homotopical interpretation of globular complex by multipointed d-space http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html

About the Cole-Ström model category structure with a locally presentable category

2013-03-07T12:44:55Z

In "Many homotopy categories are homotopy categories" (M. Cole / Topology and its Applications 153 (2006) 1084–1099), Cole generalizes the construction of the model category of Ström to any bicomplete category which is enriched, tensored and cotensored over compactly generated spaces and which satisfies an additional condition which is called the cofibration hypothesis (the mapping path object preserves colimits of towers of strong acyclic cofibrations). Now here are my questions :

1) where is exactly used in the proof the fact that the topological spaces are weak Hausdorff (i.e. for any continuous map $f:K\to X$ with $K$ compact Hausdorff, $f(K)$ is closed in $X$) ?

2) I ask the question since I would like to use this theorem for a locally presentable category which is enriched, tensored and cotensored over $\Delta$-generated spaces (i.e. colimits of simplices). Suppose that the weak Hausdorff condition above can be dropped, we then just have to check the cofibration hypothesis ; but the mapping path object turns out to be accessible so there is nothing to check.

3) If the weak Hausdorff condition is necessary in Cole's theorem, do you think that it can be replaced in the context of $\Delta$-generated spaces by the following separation condition (I don't know its name) : a $\Delta$-generated space $X$ is $\Delta$-Hausdorff if for any continuous map $f:\Delta^n \to X$, $f(\Delta^n)$ is closed in $X$.

Concerning $\Delta$-generated spaces, a short bibliography :

Notes on Delta-generated spaces by Dugger : http://math.uoregon.edu/~ddugger/delta.html
The proof that they assemble to a locally presentable category : A convenient category for directed homotopy by Fajstrup-Rosicky : http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html
A survey of their properties : Section 2 of Homotopical interpretation of globular complex by multipointed d-space http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html

About the $n$-cube

2012-10-26T19:27:04Z

$[n]$ is the set $\{0,1\}^n$ equipped with the product order $(\epsilon_1,\dots,\epsilon_n) \leq (\eta_1,\dots,\eta_n)$ if and only if $\forall i=1,\dots,n$, $\epsilon_i \leq \eta_i$. Let $$d((\epsilon_1,\dots,\epsilon_n),(\eta_1,\dots,\eta_n)) = \sum_{i=1}^{i=n}|\epsilon_i-\eta_i|.$$ A set map $f$ from $[m]$ to $[n]$ is adjacency-preserving if it is strictly increasing for the product order and if $d((\epsilon_1,\dots,\epsilon_m),(\eta_1,\dots,\eta_m)) = 1$ implies $d(f(\epsilon_1,\dots,\epsilon_m),f(\eta_1,\dots,\eta_m)) = 1$. Example : the adjacency-preserving maps from $[2]$ to itself are $(x_1,x_2)\mapsto (x_1,x_2)$, $(x_1,x_2)\mapsto (x_2,x_1)$, $(x_1,x_2)\mapsto (\min(x_1,x_2),\max(x_1,x_2))$ and $(x_1,x_2) \mapsto (\max(x_1,x_2),\min(x_1,x_2))$.

Let $n\geq 3$. Let $f:[n]\to [n]$ be an adjacency-preserving map which commutes with all automorphisms of $[n]$ (the set of automorphisms of $[n]$ is in bijection with the permutation of the set $\{1,\dots,n\}$ by permuting the coordinates). Is $f$ necessarily the identity of $[n]$ ? For $n=2$, $(x_1,x_2)\mapsto (x_2,x_1)$ is a counter-example.

PS : the definitions are in this paper http://www.pps.univ-paris-diderot.fr/~gaucher/symcub.pdf (published paper here http://dx.doi.org/10.1016/j.tcs.2009.11.013 if you have access);

Answer by Philippe Gaucher for Most striking applications of category theory?

2012-07-28T09:05:19Z

The recent developments in homotopical algebra (after 1990) would not be possible without the use of category theory, and more precisely the theory of locally presentable and accessible categories. I am talking about the theory of combinatorial model categories (model categories such that the underlying category is locally presentable).

Bibliographical reference needed (characterizing the weak equivalences of a model category)

2012-07-27T15:16:19Z

I need a bibliographical reference for this fact: let $\mathcal{M}$ be a model category such that all objects are cofibrant; then the class of weak equivalences is the class of maps f such that $\mathcal{M}(f,T)/\simeq$ is a bijection for any fibrant object $T$ where $\simeq$ is the homotopy relation. I would prefer a reference in Hirschhorn's book (I have it but I cannot find where it is proved). Thanks in advance.

Left determined model structure on delta-generated topological spaces

2012-06-20T17:18:49Z

Consider the class of cofibrations of the Quillen model structure, restricted to delta-generated topological spaces (the full subcategory of topological spaces generated by the colimits of simplices). Under Vopenka's principle, a left determined model structure w.r.t. a given class of cofibrations always exists if the underlying category is locally presentable. Since delta-generated spaces are locally presentable, how could this left determined model structure w.r.t. Quillen cofibrations look like ? Any idea ?

Answer by Philippe Gaucher for Left determined model structure on delta-generated topological spaces

2012-06-28T07:06:43Z

I can answer my question now... Not only the Quillen model structure on $\Delta$-generated spaces is left determined, but also the hypothesis $\Delta$-generated can be removed. The left determined model structure exists by Marc Olschok's PhD. The Quillen model structure has the same class of cofibrations and more weak equivalences. So the left determined model structure has more fibrant objects, that is all topological spaces. So the left determined model structure and the Quillen model structure have the same class of cofibrations and the same class of fibrant objects (all topological spaces). Therefore they are equal.

Answer by Philippe Gaucher for Negative objects in categories

2012-06-20T17:53:19Z

With Yoneda ? For every object $X$, $Mor(a\oplus (-a),X)$ is a singleton since $a \oplus (-a)$ is initial. And $Mor(a\oplus (-a),X) \cong Mor(a,X) \times Mor((-a),X)$ is a singleton as well. So $Mor(a,X) \cong Mor((-a),X) \cong Mor(0,X)$. Hence $a \cong (-a) \cong 0$.

Comment by Philippe Gaucher

2013-04-30T19:00:42Z

It is yes if the category of cofibrations is accessible... I have no more information.

Comment by Philippe Gaucher

2013-04-30T17:13:53Z

I believed that "On combinatorial model categories" by J. Rosicky was the answer (math.muni.cz/~rosicky/papers/comb2.pdf), and page 8 (top of the page), one can read that $cof(S)$ is not always accessible, $S$ being a set ! On the contrary, $inj(S)$ is always accessible (Proposition 3.3 of the same paper) as a small injectivity class...

Comment by Philippe Gaucher

2013-04-30T16:01:44Z

I cannot edit my comment so I rephrase my question. Take a "very good" cofibrant replacement $f.g$ of $X$ ($f$ cofibration and $g$ trivial fibration). Why does it come from a factorization by a functor $T$ constructed using the small object argument ?

Comment by Philippe Gaucher

2013-04-30T15:56:37Z

Indeed, the class of accessible categories is closed under lax limits, not under limits. I do understand that your homotopy pullback proves that the class of composables maps (f,g) with $f.g=\varnothing \to X$ is accessible. But the intersection with the image of the functor $T$ does not answer the question. How do you choose $T$ ? Where does it come from ?

Comment by Philippe Gaucher

2013-04-30T06:34:33Z

There is a similar (but not sure that it is simpler) problem. In the category of $\Delta$-generated spaces, is the class of cofibrant contractible spaces accessible ? My intuition tells me "yes".

Comment by Philippe Gaucher

2013-04-30T06:12:48Z

I don't understand your proof. Do you mean the map $\mathcal{C}^{[2]}\to \mathcal{C}^{[1]}$ which takes the composite ? And even with that, I still do not understand. By cofibrant replacement of $X$, I mean a pair $(X,f:Y\to X)$ where $Y$ is cofibrant and $f$ is a weak equivalence (I am also interested in the more restricted definition $f$ trivial fibration). And why use homotopy limits ? I believe (but I may be wrong) that the class of accessible categories, unlike locally presentable ones, is closed under a lot of operations like limits.

Comment by Philippe Gaucher

2013-04-29T10:39:22Z

I added the reason in my question.

Comment by Philippe Gaucher

2013-03-07T15:22:06Z

The paper you mention even gives the answer: top of page 24 : "We should remark that any locally presentable topologically bicomplete category also satisfies our hypothesis". And as far as I can understand the paper, any convenient category of topological spaces is fine, by convenient it is meant cartesian closed and containing enough topological spaces (e.g. CW-complexes).

Comment by Philippe Gaucher

2012-12-18T15:49:37Z

Here is the beginning of an idea: by right-Bousfield localizing by all trivial fibrations, you will reduce the class of cofibrations. So the trivial fibrations must be interpreted as colocal equivalences. So I suggest first a right Bousfield localization by the set of simplices, and then if the new model category has exactly the monomorphisms as cofibrations (?), it should be "between" the minimal model structure and the usual model structure by Cisinski's result (so it should be left proper), then a left Bousfield localization by the accessible class of weak equivalences could work.

Comment by Philippe Gaucher

2012-07-31T04:14:50Z

It is a shortcut for denoting the map $\mathcal{M}(Y,T)/\simeq \rightarrow \mathcal{M}(X,T)/\simeq$ if $f:X\rightarrow Y$.

Comment by Philippe Gaucher

2012-06-28T21:53:06Z

Actually, there is a slightly shorter argument: Marc Olschok already proves in his PhD that, in this case, all objects of the left determined one are fibrant. So the left determined one has the same cofibration and the same fibrant objects than Quillen's, so the two model structures are equal.

Comment by Philippe Gaucher

2012-06-28T21:48:21Z

I mean: every trivial cofibration of the left determined one is a trivial cofibration of Quillen's, so dually, every fibrant of the Quillen's is fibrant in the left determined one, i.e. all spaces are fibrant in the left determined one. BTW, the same argument also proves that with Vopenka's principle, every combinatorial model category such that all objects are fibrant is left determined, fact that I had never noticed before asking this question.

Comment by Philippe Gaucher

2012-06-22T18:00:44Z

Maybe I should say that I asked this question because I am trying to understand what the homotopy exchange property means for topological spaces. This notion is introduced by Marc Olschok in his PhD to generalize Lafont-Metayer-Worytkiewicz's construction of the folk model structures on globular $\omega$-categories. Marc told me that the homotopy exchange property is satisfied by the cylinder of topological spaces. So unless I am missing something, that implies that the Quillen model structure on delta-generated spaces is left determined, which is very weird.

Comment by Philippe Gaucher

2012-06-20T18:05:20Z

Yes, the only hypothesis is $a \oplus (-a)$ initial.