Abstract
To use of colimits to put structures together is not uncommon inmathematics; coproducts particularly of structures such as groups and vectorspaces have been known for a long time. Colimits have also been used incomputer science for example to put together labeled graphs, in system theoryetc. The importance of category for theoretical computer scientists is everydayincreasing.…nite (co)limits in the category of quadratic modules of groups by carefulconstruction of (co)product ob ject and (co)equilaser of morphisms of quadraticmodules. Moreover, we give some examples of coproduct
Highlights
Crossed modules were initially de...ned by Whitehead [12] as an algebraic model for homotopy connected 2-types
To use of colimits to put structures together is not uncommon in mathematics; coproducts of structures such as groups and vector spaces have been known for a long time
In order to contribute usage of colimits, we show existence of ...nitelimits in the category of quadratic modules of groups by careful construction ofproduct object andequilaser of morphisms of quadratic modules
Summary
Crossed modules were initially de...ned by Whitehead [12] as an algebraic model for homotopy connected 2-types. Nizar [9] has shown the existence of ...nite limits and colimits in the category of crossed R-modules of commutative algebras. Baues [2] de...ned the notion of a quadratic module of groups as an algebraic model for homotopy connected 3-types and gave a relation between quadratic modules and simplicial groups. In [1] Arvasi and Ulualan have explored relations among some algebraic models for homotopy 3-types such as quadratic modules, 2-crossed module,crossed square and simplicial groups. We show the existence of ...nite colimits of category of quadratic modules. The following proposition [5] leads us to ...nd ...nite colimit in the category of quadratic modules. We show the existence of ...nite limits and colimits in the category of quadratic R-modules
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