The Category of Matroids
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The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful and having a nearly full Kan extension; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial. Finally, a characterization of matroids in terms of optimality of the greedy algorithm can be rephrased in terms of limits.
- 1.Abramsky, S.: Abstract scalars, loops, and free traced and strongly compact closed categories Algebra and Coalgebra in Computer Science, pp. 1–29. Springer (2005)Google Scholar
- 2.Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and concrete categories. John Wiley (1990)Google Scholar
- 3.Al-Hawary, T.A.: Toward an elementary axiomatic theory of the category of loopless pointed matroids and pointed strong maps. Ph.D. thesis, University of Montana (1997)Google Scholar
- 6.Anderson, L.: Matroid bundles. New perspectives in algebraic combinatorics, 1–21 (1999)Google Scholar
- 10.Crapo, H.: Constructions in combinatorial geometries. In: NSF Advanced Science Seminar in Combinatorial Theory)(Notes, Bowdoin College) (1971)Google Scholar
- 11.Heunen, C.: Semimodule enrichment. In: Mathematical Foundations of Programming Semantics XXIV, no. 218 in Electronic Notes in Theoretical Computer Science, pp. 193–208 (2008)Google Scholar
- 16.Kung, J.P.S.: Twelve views of matroid theory. Combinatorial and Computational Mathematics: Present and Future (2001)Google Scholar
- 19.Mac Lane, S.: Categories for the working mathematician, 2nd edn. Springer (1971)Google Scholar
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