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| In [[algebraic topology]], a discipline within [[mathematics]], the '''acyclic models theorem''' can be used to show that two [[homology theories]] are [[isomorphic]]. The [[theorem]] was developed by topologists [[Samuel Eilenberg]] and [[Saunders MacLane]]. They discovered that, when topologists were writing proofs to establish equivalence of various homology theories, there were numerous similarities in the processes. Eilenberg and MacLane then discovered the theorem to generalize this process.
| | Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. What I advise you do next is save the backup data file to a remote place like a CD-ROM, external disk drive if you have one or a provider such as Dropbox. The effect is to promote older posts by moving them back onto the front page and into the rss feed. Donor oocytes and menopausal pregnancy: Oocyte donation to women of advanced reproductive age: pregnancy results and obstetrical outcomes in patients 45 years and older. Understanding how Word - Press works can be a challenge, but it is not too difficult when you learn more about it. <br><br>Always remember that an effective linkwheel strategy strives to answer all the demands of popular search engines while reacting to the latest marketing number trends. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. Some plugins ask users to match pictures or add numbers, and although effective, they appear unprofessional and unnecessary. So, if you are looking for some option to build a giant e-commerce website, then e-shopping preferable CMS tools will be helpful for you. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience. <br><br>Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. As of now, Pin2Press is getting ready to hit the market. In case you loved this informative article and you want to receive much more information concerning [http://www.addbabes.com/crtr/cgi/out.cgi?u=https://wordpress.org/plugins/ready-backup/ backup plugin] kindly visit the website. This platform can be customizedaccording to the requirements of the business. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait. <br><br>Word - Press installation is very easy and hassle free. But the Joomla was created as the CMS over years of hard work. Websites that do rank highly, do so becaue they use keyword-heavy post titles. A whole lot worse, your site will likely be useless as well as your merchandise won't sell if no one has the endurance to wait for the web pages to load. If your site does well you can get paid professional designer to create a unique Word - Press theme. <br><br>As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. The 2010 voting took place from July 7 through August 31, 2010. |
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| It can be used to prove the [[Eilenberg–Zilber theorem]].
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| ==Statement of the theorem==
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| Let <math>\mathcal{K}</math> be an arbitrary [[category (mathematics)|category]] and <math>\mathcal{C}(R)</math> be the category of chain complexes of <math>R</math>-[[Module (mathematics)|module]]s. Let <math>F,V : \mathcal{K} \to \mathcal{C}(R)</math> be [[covariant functor]]s such that:
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| * <math> F_i = V_i = 0 </math> for <math> i < 0</math>.
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| * There are <math>\mathcal{M}_k \subseteq \mathcal{K}</math> for <math>k \ge 0</math> such that <math>F_k</math> has a basis in <math> \mathcal{M}_k </math>, so <math>F</math> is a [[free functor]].
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| * <math>V</math> is <math>k</math>- and <math>(k+1)</math>-acyclic at these models, which means that <math>H_k(V(M)) = 0</math> for all <math>k>0</math> and all <math>M \in \mathcal{M}_k \cup \mathcal{M}_{k+1}</math>.
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| Then the following assertions hold:
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| * Every [[natural transformation]] <math>\varphi : H_0(F) \to H_0(V)</math> is induced by a natural chain map <math>f : F \to V</math>.
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| * If <math>\varphi,\psi: H_0(F)\to H_0(V)</math> are natural transformations, <math>f,g: F\to V</math> are natural chain maps as before and <math>\varphi^{M}=\psi^{M}</math> for all models <math>M\in\mathcal{M}_0</math>, then there is a natural chain homotopy between <math>f</math> and <math>g</math>.
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| * In particular the chain map <math> f</math> is unique up to natural [[chain homotopy]].<ref> {{Citation | last1=Dold | first1=Albrecht | title=Lectures on Algebraic Topology | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=A Series of Comprehensive Studies in Mathematics | isbn= 3-540-10369-4 | edition=2nd | year=1980 | volume=200}}</ref>
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| == Generalizations ==
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| === Projective and acyclic complexes ===
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| What is above is one of the earliest versions of the theorem. Another
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| version is the one that says that if <math>K</math> is a complex of
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| projectives in an [[abelian category]] and <math>L</math> is an acyclic
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| complex in that category, then any map <math>K_0 \to
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| L_0</math> extends to a chain map <math>K\to L</math>, unique up to
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| homotopy.
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| This specializes almost to the above theorem if one uses the functor category <math>\mathcal{C}(R)^\mathcal{K}</math> as the abelian category. Free functors are projective objects in that category. The morphisms in the functor category are natural transformations, so the constructed chain maps and homotopies are all natural. The difference is that in the above version, <math>V</math> being acyclic is a stronger assumption than being acyclic only at certain objects.
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| On the other hand, the above version almost implies this version by letting <math>\mathcal{K}</math> a category with only one object. Then the free functor <math>F</math> is basically just free (and hence projective) module. <math>V</math> being acyclic at the models (there is only one) means nothing else than that the complex <math>V</math> is acyclic.
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| === Acyclic classes ===
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| Then there is the grand theorem that unifies them all. Let <math>
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| \mathcal{A}</math> be an abelian category (for example <math>\mathcal{C}(R)</math> or <math>\mathcal{C}(R)^\mathcal{K}</math>). A class <math>\Gamma</math> of chain complexes over <math>\mathcal{A}</math> will be called an '''acyclic class''' provided:
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| * The 0 complex is in <math>\Gamma</math>.
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| * The complex <math>C</math> belongs to <math>\Gamma</math> if and only if the suspension of <math>C</math> does.
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| * If the complexes <math>K</math> and <math>L</math> are homotopic and <math>K \in\Gamma</math>, then <math>L\in\Gamma</math>.
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| * Every complex in <math>\Gamma</math> is acyclic.
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| * If <math>D</math> is a double complex, all of whose rows are in <math>\Gamma</math>, then the total complex of <math>D</math> belongs to <math>\Gamma</math>.
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| There are three natural examples of acyclic classes, although doubtless
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| others exist. The first is that of homotopy contractible complexes.
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| The second is that of acyclic complexes. In functor categories (e.g. the
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| category of all functors from topological spaces to abelian groups),
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| there is a class of complexes that are contractible on each object, but
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| where the contractions might not be given by natural transformations.
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| Another example is again in functor categories but this time the complexes are acyclic only at certain objects.
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| Let <math>\Sigma</math> denote the class of chain maps between complexes
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| whose mapping cone belongs to <math>\Gamma</math>. Although
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| <math>\Sigma</math> does not necessarily have a calculus of either right
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| or left fractions, it has weaker properties of having homotopy classes
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| of both left and right fractions that permit forming the class
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| <math>\Sigma^{-1} C</math> gotten by inverting the arrows in
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| <math>\Sigma</math>.{{Citation needed|date=February 2010}}
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| Let <math>G</math> be an augmented endofunctor on <math>C</math>,
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| meaning there is given a natural transformation
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| <math>\epsilon:G\to Id</math> (the identity functor on <math>C</math>). We say that the chain complex <math>K</math> is <math>G</math>-''presentable'' if for each <math>n</math>, the chain
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| complex
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| :<math>\cdots K_nG^{m+1}\to K_nG^{m}\to \cdots \to K_n</math>
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| belongs to <math>\Gamma</math>. The boundary operator is given by
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| :<math>\sum (-1)^i K_nG^i\epsilon G^{m-i}:K_nG^{m+1}\to K_nG^m</math>.
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| We say that the chain complex functor <math>L</math> is
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| <math>G</math>-''acyclic'' if the augmented chain complex
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| <math>L\to H_0(L)\to 0</math> belongs to <math>\Gamma</math>.
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| '''Theorem'''. ''Let <math>\Gamma</math> be an acyclic class and
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| <math>\Sigma</math> the corresponding class of arrows in the category of
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| chain complexes. Suppose that <math>K</math> is <math>G</math>-presentable and <math>L</math> is <math>G</math>-acyclic.
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| Then any natural transformation <math>f_0:H_0(K)\to H_0(L)</math>
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| extends, in the category <math>\Sigma^{-1}(C)</math> to a natural
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| transformation of chain functors <math>f:K\to L</math> and this is
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| unique in <math>\Sigma^{-1}(C)</math> up to chain homotopies.
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| If we suppose, in addition, that <math>L</math> is <math>G</math>-presentable, that <math>K</math> is <math>G</math>-acyclic, and that <math>f_0</math> is an isomorphism, then <math>f</math> is homotopy equivalence.
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| == Example ==
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| Here is an example of this last theorem in action. Let <math>X</math>
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| be the category of triangulable spaces and <math>C</math> be the
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| category of abelian group valued functors on <math>X</math>. Let
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| <math>K</math> be the singular chain complex functor and <math>L</math>
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| be the simplicial chain complex functor. Let <math>E: X\to | |
| X</math> be the functor that assigns to each space <math>X</math> the
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| space <math>\sum_{n\ge 0}\sum_{\textrm{Hom}(\Delta_n,X)}\Delta_n</math>. Here, <math>\Delta_n</math> is the <math>n</math>-simplex and this functor assigns to <math>X</math> the sum of as many copies of each
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| <math>n</math>-simplex as there are maps <math>\Delta_n\to X</math>.
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| Then let <math>G</math> be defined by <math>G(C)=CE</math>. There is an
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| obvious augmentation <math>EX\to X</math> and this induces one on
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| <math>G</math>. It can be shown that both <math>K</math> and
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| <math>L</math> are both <math>G</math>-presentable and
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| <math>G</math>-acyclic (the proof that <math>L</math> is not entirely
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| straigtforward and uses a detour through simplicial subdivision, which
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| can also be handled using the above theorem). The class <math>\Gamma</math> is the class of homology equivalences. It is rather
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| obvious that <math>H_0(K)\simeq H_0(L)</math> and so we conclude that
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| singular and simplicial homology are isomorphic on <math>X</math>.
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| There are many other examples in both algebra and topology, some of
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| which are described in M. Barr, Acyclic Models. AMS, 2002.
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| ==References==
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| <references/>
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| * Schon, R. Acyclic models and excision. _Proc. Amer. Math. Soc._ 59~(1) (1976) 167--168.
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| [[Category:Homological algebra]]
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| [[Category:Theorems in algebraic topology]]
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| {{DEFAULTSORT:Acyclic Model}}
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Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. What I advise you do next is save the backup data file to a remote place like a CD-ROM, external disk drive if you have one or a provider such as Dropbox. The effect is to promote older posts by moving them back onto the front page and into the rss feed. Donor oocytes and menopausal pregnancy: Oocyte donation to women of advanced reproductive age: pregnancy results and obstetrical outcomes in patients 45 years and older. Understanding how Word - Press works can be a challenge, but it is not too difficult when you learn more about it.
Always remember that an effective linkwheel strategy strives to answer all the demands of popular search engines while reacting to the latest marketing number trends. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. Some plugins ask users to match pictures or add numbers, and although effective, they appear unprofessional and unnecessary. So, if you are looking for some option to build a giant e-commerce website, then e-shopping preferable CMS tools will be helpful for you. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience.
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As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. The 2010 voting took place from July 7 through August 31, 2010.