Quadratic modules bifibred over nil(2)-modules

Hasan Atik; Erdal Ulualan

doi:10.1007/s40062-015-0122-y

Journal of Homotopy and Related Structures

March 2017, Volume 12, Issue 1, pp 83–108 | Cite as

Quadratic modules bifibred over nil(2)-modules

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Authors and affiliations

Hasan Atik
Erdal UlualanEmail author

Article

First Online: 10 December 2015

Received: 23 September 2015

Accepted: 10 November 2015

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Abstract

In this work, we show that the forgetful functor from the category of Baues’s quadratic modules to that of nil(2)-modules is a bifibration.

Keywords

Crossed Modules Quadratic Modules Fibration and Cofibration of Categories

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Notes

Acknowledgments

We would like to thank the referee for helpful comments and improvements to the paper.

Appendix

The Proof of Proposition 4.2

$(\mathsf {[A]})$ We show that the following diagram

Open image in new window

is a quadratic module.

$\mathsf {QM1}$. It is already known that $\partial ':R\rightarrow S$ is a nil(2)-module and, since $r\in \ker \partial '_1$, we obtain $\partial _1^{\prime }\partial _2^{\prime }(r,l)=\partial _1^{\prime }(r)=1.$ That is,

Open image in new window

is a complex of groups.

$\mathsf {QM2}$. For all $r,r^{\prime }\in R$, we obtain

$$\begin{aligned} \partial _2^{\prime }\omega ^{\prime }(\{r\}\otimes \{r^{\prime }\})=\partial _2^{\prime }(\langle r,r^{\prime }\rangle , \omega \{u_1r\}\otimes \{u_1r'\})=\langle r,r'\rangle =w'(\{r\}\otimes \{r'\}). \end{aligned}$$

$\mathsf {QM3}$. For $(r,l)\in u^*(L)$, $r^{\prime }\in R$, we obtain

$$\begin{aligned}&\omega ^{\prime }(\{\partial _2^{\prime }(r,l)\}\otimes \{(r^{\prime })\}\{r^{\prime }\}\otimes \{\partial _2^{\prime }(r,l)\})\\&\quad =\omega ^{\prime }(\{r\}\otimes \{r'\}\{r'\}\otimes \{r\})\\&\quad =( \langle r,r'\rangle ,\omega \{u_1r\}\otimes \{u_1r' \})( \langle r',r\rangle ,\omega \{u_1r'\}\otimes \{u_1r\}) \\&\quad =\left( (r^{-1}r^{\prime -1}rr^{\prime \partial _1^{\prime }r})((r')^{-1}r^{-1}(r') r^{\partial _1'r'}),\omega \{\partial _2 l\}\otimes \{u_1r' \}\{u_1r'\}\otimes \{\partial _2 l\}\right) \\&\quad =\left( (r^{-1}r^{\prime -1}rr^{\prime \partial _1^{\prime }r})((r')^{-1}r^{-1}(r') r^{\partial _1'r'}), l^{-1}l^{\partial _1 u_1(r')}\right) (\text {since }u_1(r)=\partial _2l) \\&\quad =(r^{-1}r^{\partial _1^{\prime }r^{\prime }},l^{-1}l^{\partial _1 u_1(r')})\quad (\text {since } r\in \ker \partial _1') \\&\quad =(r,l)^{-1}(r^{\partial _1'(r')},l^{u_0\partial _1'r'})\\&\quad =(r,l)^{-1}(r,l)^{\partial _1'r'} \end{aligned}$$

$\mathsf {QM4.}$ For $(r,l), (r^{\prime },l^{\prime })\in u^*(L)$, we get

$$\begin{aligned} \omega ^{\prime }(\{\partial _2^{\prime }(r,l)\}\otimes \{\partial _2^{\prime }(r,l)\})&=( \langle r,r^{\prime }\rangle ,\omega \{u_1r\}\otimes \{u_1r^{\prime }\}) \\&=(r^{-1}r^{\prime -1}rr^{\prime \partial _1^{\prime }r},\omega \{\partial _2l\}\otimes \{\partial _2l^{\prime }\}) \\&=(r^{-1}r^{\prime -1}rr^{\prime },[l,l^{\prime }])\quad (\text {since } r\in \ker \partial _1') \\&=([r,r^{\prime }],[l,l^{\prime }]) \\&=[(r,l),(r^{\prime },l^{\prime })]. \end{aligned}$$

$(\mathsf {[B]})$ We show that $(\varphi ,u_1,u_0)$ is a map of quadratic modules. Since

$$\begin{aligned} u_1(\upsilon _1\partial ''_2(z))=u'_1(\partial ''_2 z)=\partial _2(\theta (z)) \end{aligned}$$

and $\partial _1^{\prime }(\upsilon _1\partial _2^{\prime \prime }z)=\upsilon _0\partial _1^{\prime \prime }\partial _2^{\prime \prime }z=1$, we have $\upsilon _1\partial ''_2(z)\in \ker \partial '_1$, and $(\upsilon _1\partial ''_2(z),\theta (z))\in u^*(L)$. We obtain

$$\begin{aligned} \partial '_2\psi (z)=\partial '_2(\upsilon _1\partial ''_2(z),\theta z)=\upsilon _1\partial ''_2(z) \end{aligned}$$

for $z\in Z$ and

$$\begin{aligned} \psi (\omega ''\{x\}\otimes \{y\})&=(\upsilon _1(\partial ''_2\omega ''\{x\}\otimes \{y\}),\theta (\omega ''\{x\}\otimes \{y\}))\\&=(\upsilon _1(\langle x,y\rangle ),\omega \{u'_1(x)\}\otimes \{u'_1(y)\})\\&=(\langle \upsilon _1x,\upsilon _1y\rangle , \omega \{u_1(\upsilon _1x)\}\otimes \{u_1(\upsilon _1y)\})\\&=\omega '(\{\upsilon _1x\}\otimes \{\upsilon _1y\}) \end{aligned}$$

for $\{x\}\otimes \{y\}\in C''\otimes C'' $. $\square $

The Proof of Proposition 5.1

$(\mathsf {[C]})$ We have for $(n,l)\in F(N\times L)$,

$$\begin{aligned} \partial '_1(\partial '_2)(n,l)&=\partial '_1(n^{-1}v_1\partial _2(l)n)\\&=\partial '_1(n^{-1})\partial '_1 v_1\partial _2(l)\partial '_1n\\&=\partial '_1(n^{-1}) v_0(\partial _1\partial _2l)\partial '_1n \quad (\text {since } \partial '_1v_1=v_0\partial _1)\\&=\partial '_1(n^{-1})v_0(1)\partial '_1n \quad (\text {since } \partial _1\partial _2l=1)\\&=1 \end{aligned}$$

and for $\langle \{x\}\otimes \{y\}\rangle \in \langle C\otimes C\rangle $,

$$\begin{aligned} \partial '_1(\partial '_2 \langle \{x\}\otimes \{y\}\rangle )=\partial '_1(x^{-1}y^{-1}xy^{\partial '_1x}) =1. \end{aligned}$$

Thus the diagram

Open image in new window

is a complex of groups.

$(\mathsf {[D]})$ We show that $(\psi ,v_1,v_0)$ is a morphism of quadratic modules. We obtain $\partial '_2\psi (l)=\partial '_2(1,l)=v_1\partial _2(l)$ for $l\in L$ and

$$\begin{aligned} \psi (\omega \{x\}\otimes \{y\})=(1,\omega \{x\}\otimes \{y\})=\langle \{v_1(x)\}\otimes \{v_1(y)\} \rangle =\omega '(\{v_1(x)\}\otimes \{v_1(y)\}) \end{aligned}$$

for $x,y\in M$ and

$$\begin{aligned} \psi (l^p)=(1,l^p)=(v_0p,l)=(1,l)^{v_0p}=\psi (l)^{v_0p} \end{aligned}$$

for $p\in P$.

$(\mathsf {[E]})$ We show that $(\theta _*,id,id)$ is a morphism of quadratic modules. We obtain for $n\in N$ and $l\in L$

$$\begin{aligned} \theta _{*}((n,l)^{q})=\theta _{*}(n^q,l)=(\theta '(l))^{n^{q}}=(\theta '(l)^{n})^{id(q)}=(\theta _*(n,l))^{id(q)} \end{aligned}$$

and for $x,y\in N$,

$$\begin{aligned}&\theta _{*}(\langle \{x\}\otimes \{y\}\rangle ^{q^{\prime }})=\theta _{*}\langle \{x^{q^{\prime }}\}\otimes \{y^{q^{\prime }}\}\rangle =\omega ^{\prime \prime }(\{x^{q^{\prime }}\}\otimes \{y^{q^{\prime }}\})=\omega ^{\prime \prime }(\{x\}\otimes \{y\})^{q^{\prime }}\\&\quad =(\theta _{*}\langle \{x\}\otimes \{y\}\rangle )^{id(q^{\prime })} \end{aligned}$$

Further, we obtain

$$\begin{aligned} \partial ''_2(\theta _*(n,l))&=\partial ''_2(\theta '(l)^n)\\&=n^{-1}(\partial ''_2(\theta '(l)))n\\&=n^{-1}(v_1\partial _2(l))n\\&=id(\partial '_2(n,l)) \end{aligned}$$

and

$$\begin{aligned} \partial ''_2(\theta _*\langle \{x\}\otimes \{y\}\rangle )&=\partial ''_2(\omega ''\{x\}\otimes \{y\})\\&=w(\{x\}\otimes \{y\})\\&=x^{-1}y^{-1}xy^{\partial '_1x}\\&=\partial '_2\langle \{x\}\otimes \{y\}\rangle \end{aligned}$$

for $(n,l)\in F(N\times L)$ and $\langle \{x\}\otimes \{y\}\rangle \in \langle C\otimes C\rangle $. Therefore, $(\theta _{*},id,id)$ is the unique morphism in $\mathsf {Quad}_{(N\rightarrow Q)}$. $\square $

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Authors and Affiliations

Hasan Atik
- 1
Erdal Ulualan
- 2
Email author

1.Mathematics Department, Science Facultyİstanbul Medeniyet UniversityIstanbulTurkey
2.Mathematics Department, Science and Art FacultyDumlupınar UniversityKutahyaTurkey

Quadratic modules bifibred over nil(2)-modules

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