User fedotov - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T13:46:47Z http://mathoverflow.net/feeds/user/21369 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121453/mapping-space-between-classifying-spaces mapping space between classifying spaces Fedotov 2013-02-11T09:13:46Z 2013-02-12T03:06:53Z <p>I wanted to ask a summary of known results and references about the homotopy type of the mapping space $\mathrm{Map}(BG,BK)$ (and specially the connected components) between the classifying spaces when G and K are general topological groups. Thank you. </p> http://mathoverflow.net/questions/114553/loop-space-of-a-category/115502#115502 Answer by Fedotov for Loop space of a category Fedotov 2012-12-05T14:11:55Z 2012-12-06T07:16:23Z <p>I will try to answer the question. As I said in a comment, the Thomason model structure on $Cat$ is not simplicial model structure. Let $C$ be a small category, we will view it as a topological category. Denote by $C[C^{-1}]$ the topological category where we invert all maps of $C$ such that $C\rightarrow C[C^{-1}]$ is a cofibration of topological categories, then the coherent nerve $N_{\bullet}C\rightarrow N_{\bullet}C[C^{-1}]$ induces a weak equivalence of simplicial sets. Notice that $C[C^{-1}]$ is an infinity groupoid. </p> <p>Let $C$ be a cofibrant topological category. The mapping space $map(C,D)$ in the model category of topological categories is given by the (standard) nerve of the following $HOM(C,D)$ category :</p> <p>$\underline{Objects}$ are topological functor $F:C^{op}\times D:\rightarrow Top$ such that for any $c\in C,$ $F(c)$ is equivalent to a representable functor $D(d,-)$ for some $d\in D$. </p> <p>$\underline{Morphisms}$ in this category are natural transformation $H:F\rightarrow G$ such that $F(c,d)\rightarrow G(c,d)$ is a weak equivalence for all $c\in C$ and $d\in D$.</p> <p>Let $S^{1}$ a simplicial model for a circle. Let $k: sSet\rightarrow sSet$ the cocontinues Joyal functor which take $\Delta^{n}$ to the nerve of the groupoid with $n+1$ objects and only one isomorphism between any two objects.</p> <p>Recall that $\mathfrak{C}: sSet\rightarrow Cat_{\Delta}$ is the left quillen adjoint to the coherent $N_{\bullet}$ betwen the joyal model structre on $sSet$ and the Bergner model structure on $Cat_{\Delta}$ </p> <p>Now $k(S^{1})$ is a simplicial set, and the cofibrant topological category $|\mathfrak{C}[k(S^{1})]|$ is an infinity groupoid and its cohenrent nerve is equivalent to $S^{1}$.<br> The finial result is that $HOM(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])$ is a model for $\Lambda C$, since the nerve of $HOM(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])$ is equivalent to $map(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])\sim Map(S^{1},N_{\bullet}C[C^{-1}])\sim Map(S^{1}, N_{\bullet} C)=\Lambda N_{\bullet}C$ </p> <p>N.B. The only point that I did not explained is the construction of $C[C^{-1}]$. Let 1 be the category with two objets a and b and a unique morphisms from $: f:a\rightarrow b$. Let $\widehat{1}=|\mathfrak{C}[k(\Delta^{1})]|$, then $C[C^{-1}]$ is the pushout $colim (\sqcup_{mor C} \widehat{1}\leftarrow \sqcup_{mor C} 1\rightarrow C )$ i.e., for each porphism of $C$ there is a map $1\rightarrow C$. </p> http://mathoverflow.net/questions/113853/string-topology-for-a-lie-group String topology for a Lie group Fedotov 2012-11-19T18:14:37Z 2012-11-27T00:48:57Z <p>My question is very naive maybe, I don't have a deep knowledge about string topology. I wanted to ask (explanation or a reference) for the geometric interpretation of free loop space (continues maps) $\mathrm{Map}(S^{1}, G)$, when G is a topological group (Lie group). Does it classify something geometric? </p> <p>Thank you.</p> http://mathoverflow.net/questions/110658/fubini-theorem-for-hocolim Fubini theorem for hocolim. Fedotov 2012-10-25T13:50:29Z 2012-10-26T12:05:06Z <p>I wanted to ask the following question, Suppose $\mathbf{M}$ a cof generated model category and $I,~J$ two small categories. Suppose that $F:J\rightarrow \mathbf{M}^{\mathrm{I}}$ is a functor. Is it true that the map $\mathrm{hocolim}_{J}(F(i))\rightarrow (\mathrm{hocolim}_J~F)(i)$ is a weak equivalence in $\mathbf{M}$ ?</p> <p>For more precision: $F(i)$ is the evaluation of the functor $F:J\times I\rightarrow \mathbf{M}$ at $i\in I$, and $\mathrm{hocolim}_J~F$ is an object in the model category $\mathbf{M}^{I}$. The categories $\mathbf{M}^{I}$, $\mathbf{M}^{J\times I}$ are equipped with the projective model structure.</p> <p>Thank you.</p> http://mathoverflow.net/questions/92455/cartesian-and-strictly-associative-path-object Cartesian and strictly associative Path Object. Fedotov 2012-03-28T13:25:24Z 2012-03-28T13:25:24Z <p>Let $X$ be a topological space. Is there a path object functor $\mathrm{P}:\mathbf{Top}\rightarrow \mathbf{Top}$, with fibration $ev_{0},ev_{1}: \mathrm{P}(X)\rightarrow X\times X$ such that: </p> <p>(1) $\mathrm{P}$ is a cartesian functor i.e., $\mathrm{P}(X\times Y)$ is isomorphic to $\mathrm{P}(X)\times \mathrm{P}(Y).$ </p> <p>(2) and the concatenation map $-\ast-:\mathrm{P}(X)\times_{X}\mathrm{P}(X)\rightarrow \mathrm{P}(X)$ is strictly associative. </p> <p>The condition (1) (but not (2)) is verified by the standard path object $\mathrm{P}(-)= \mathrm{Map}([0,1],-)$ and the condition (2) (but not (1)) is verified by the moore path object $\mathrm{M}(-)$ which is not a cartesian functor. </p> http://mathoverflow.net/questions/90480/simplicial-approximation-for-simplicial-spaces/90496#90496 Answer by Fedotov for Simplicial approximation for simplicial spaces Fedotov 2012-03-07T19:16:10Z 2012-03-07T19:16:10Z <p>I'm just reformulating your question in simplicial case. If you consider the category of simplicial sets $\mathbf{sSet}$ you can formulate your question as follows: Is the diagonal functor $diag: [\mathbf{\Delta}^{op},\mathbf{sSet}]\rightarrow \mathbf{sSet}$ from bisimplicial sets to simplicial sets homotopicaly full ? The diagonal functor is actually the realization functor. If we put the diagonal model structure on $[\mathbf{\Delta}^{op},\mathbf{sSet}]$ then $diag$ is a Quillen equivalence. As a consequence, if $f: diag(X_{\bullet, \bullet})\rightarrow diag(Y_{\bullet, \bullet})$ is a bisimplicial map and $X_{\bullet, \bullet}$, $Y_{\bullet, \bullet}$ are fibrant-cofibrant bisimplicial sets (in the diagonal model structure), then there exists $g: X_{\bullet, \bullet}\rightarrow Y_{\bullet, \bullet}$ such that $diag(g)$ is homotopic to $f$. </p> http://mathoverflow.net/questions/89900/moore-path-space Moore path space. Fedotov 2012-02-29T20:21:10Z 2012-03-01T09:04:32Z <p>Let $X$ a topological space and $MX$ the Moore path space of $X$ there is two maps from $\alpha,\omega: MX\rightarrow X$ (evaluation in 0 and evaluation at the total length). The classical path object $X^{I}$ is a subspace of $MX$. Is is true that $(\alpha,\omega): MX\rightarrow X\times X$ is a fibration and the inclusion $X^{I}\rightarrow MX$ a weak equivalence? </p> http://mathoverflow.net/questions/88354/monadicity-theorem-in-homotopy-theory Monadicity theorem in homotopy theory. Fedotov 2012-02-13T16:23:00Z 2012-02-15T05:14:53Z <p>Let $\mathbf{C}$ be a cofibrantly generated model category (assume for simplicity that all objects are fibrant) and $\mathbf{C}^{\mathrm{T}}$ the category of $\mathrm{T}$-algebras with the induced model structure (same weak equivalences and fibrations as in the underlying model category $\mathbf{C}$). By definition, the adjuction $\mathrm{T}:\mathbf{C}\rightleftharpoons\mathbf{C}^{\mathrm{T}}: \mathrm{U}$ is monadic. How about the homotopical version, i.e, $\mathbb{L}\mathrm{T}:Ho\mathbf{C}\rightleftharpoons Ho(\mathbf{C}^{\mathrm{T}}): \mathbb{R}\mathrm{U}$</p> <p>is there any result about the "homotopical" monadicity theorem, which compares $Ho(\mathbf{C}^{\mathrm{T}})$ and $Ho(\mathbf{C})^{\mathbb{L}\mathrm{T}}$. </p> http://mathoverflow.net/questions/121453/mapping-space-between-classifying-spaces/121550#121550 Comment by Fedotov Fedotov 2013-02-12T10:17:02Z 2013-02-12T10:17:02Z Thank you Peter for the references. If I'm not wrong the examples that you gave are more related to the case of (compact) Lie groups as Ralph has noticed. http://mathoverflow.net/questions/121453/mapping-space-between-classifying-spaces/121473#121473 Comment by Fedotov Fedotov 2013-02-12T10:14:36Z 2013-02-12T10:14:36Z Thank you for the references Andreas! http://mathoverflow.net/questions/121453/mapping-space-between-classifying-spaces Comment by Fedotov Fedotov 2013-02-11T10:11:51Z 2013-02-11T10:11:51Z sure, thank you, I wanted to know a summary of all well known cases... little bit more general than compact Lie groups. I was not enough precise. http://mathoverflow.net/questions/114553/loop-space-of-a-category/115502#115502 Comment by Fedotov Fedotov 2012-12-06T07:21:55Z 2012-12-06T07:21:55Z I put the construction of $C[C^{-1}]$, it does not require zig-zags and it conceptually very easy. I agree that the constructed $\Lambda C$ is not very concrete but still you have a general model for the cotensorisation functor in $Cat$ (with Thomason model structure) by $\sSet$. http://mathoverflow.net/questions/114553/loop-space-of-a-category Comment by Fedotov Fedotov 2012-11-30T15:49:25Z 2012-11-30T15:49:25Z Do you need a concrete model? The naive constructions will fail as was mentioned by Karlo Szumilo...essentially because the Thomason model structure on Cat is not a simplicial monoidal closed model category, but still you can use the derived internal hom in Cat (wich exists) Thomason model structure... The model that you obtain for $\Lambda C$ is very big. I don't know if you need more details. http://mathoverflow.net/questions/113853/string-topology-for-a-lie-group Comment by Fedotov Fedotov 2012-11-19T21:36:36Z 2012-11-19T21:36:36Z Actually I was asking if $\mathrm{Map}(S^{1},G)$ classifies a topological-geomtric structure when $G$ is a topological group... http://mathoverflow.net/questions/113853/string-topology-for-a-lie-group/113855#113855 Comment by Fedotov Fedotov 2012-11-19T19:54:52Z 2012-11-19T19:54:52Z Thank you for the references! http://mathoverflow.net/questions/95021/homotopy-limits-of-quasi-categories Comment by Fedotov Fedotov 2012-04-24T14:55:33Z 2012-04-24T14:55:33Z May be I'm wrong, but it seems to me that it should be: $Map(N(EG),N\mathbf({\mathcal{M}}^{\circ}))^{G}$ equivalent to $N((RHom(EG,\mathcal{M})^{G})^{\circ})$, where RHom is the right derived internal Hom functor defined by B.To&#235;n for $\mathrm{Ho}(\mathbf{Cat}_{\Delta})$. http://mathoverflow.net/questions/92538/transporting-model-structures-via-adjunctions Comment by Fedotov Fedotov 2012-03-29T08:49:19Z 2012-03-29T08:49:19Z For the first question (less standard, cofibration and weak equivalence) you can check section 1.2 theorem 1.13 of <a href="http://arxiv.org/pdf/0902.3393.pdf" rel="nofollow">arxiv.org/pdf/0902.3393.pdf</a> The dual version is in <a href="http://arxiv.org/pdf/math/0206094v1.pdf" rel="nofollow">arxiv.org/pdf/math/0206094v1.pdf</a> Hope it will help you http://mathoverflow.net/questions/92455/cartesian-and-strictly-associative-path-object Comment by Fedotov Fedotov 2012-03-28T16:51:43Z 2012-03-28T16:51:43Z That is not a path object! Just to be clear a path object $\mathrm{P}(X)$ factors the diagonal $\Delta: X\rightarrow X\times X$ to $X\rightarrow \mathrm{P}(X)\rightarrow X\times X$ where the first map is a weak equivalence and the second is a fibration. http://mathoverflow.net/questions/90379/equivalence-between-e-infty-spaces-and-connective-spectra Comment by Fedotov Fedotov 2012-03-07T14:54:47Z 2012-03-07T14:54:47Z The bar construction $B$ for a $E_{\infty}$-spaces is a model for the suspension functor $\Sigma$ (of coarse in the category of $E_{\infty}$-spaces). I think it is not a left Quillen functor. I would say that we have an adjunction at the level of $\mathrm{Ho}(E_{\infty}-spaces)$ between $\Omega$ and $B\sim\Sigma$. I think that in the case of commutative topological monoids, the bar construction $B$ is a left adjoint to $\Omega$. http://mathoverflow.net/questions/89900/moore-path-space/89933#89933 Comment by Fedotov Fedotov 2012-03-01T22:00:40Z 2012-03-01T22:00:40Z Thank you for all the details! http://mathoverflow.net/questions/89900/moore-path-space/89909#89909 Comment by Fedotov Fedotov 2012-03-01T05:21:25Z 2012-03-01T05:21:25Z So it is not possible to show that is actually a honest fibration ?! http://mathoverflow.net/questions/89900/moore-path-space Comment by Fedotov Fedotov 2012-02-29T22:00:09Z 2012-02-29T22:00:09Z Thank you Mark! http://mathoverflow.net/questions/88354/monadicity-theorem-in-homotopy-theory/88375#88375 Comment by Fedotov Fedotov 2012-02-14T09:25:57Z 2012-02-14T09:25:57Z Mike, Noel, Thank you for your answers!