The Category of Matroids

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]  Matthew T. Stamps.  Topological representations of matroid maps , 2011, 1104.4152.

[2]  Ronald Brown,et al.  Graphs of Morphisms of Graphs , 2008, Electron. J. Comb..

[3]  Free Quillen Factorization Systems , 2002 .

[4]  J. Kahn,et al.  Varieties of combinatorial geometries , 1982 .

[5]  Chris Heunen,et al.  On the Functor ℓ2 , 2010, Computation, Logic, Games, and Quantum Foundations.

[6]  András Recski,et al.  Elementary strong maps of graphic matroids, II , 1994, Graphs Comb..

[7]  Tom Brylawski,et al.  A combinatorial model for series-parallel networks , 1971 .

[8]  Thomas Zaslavsky,et al.  Biased graphs IV: Geometrical realizations , 2003, J. Comb. Theory, Ser. B.

[9]  Ling-Xia Lu,et al.  CATEGORICAL RELATIONS AMONG MATROIDS, FUZZY MATROIDS AND FUZZIFYING MATROIDS , 2010 .

[10]  H. Whitney.  On the Abstract Properties of Linear Dependence , 1935 .

[11]  D. Higgs,et al.  Strong maps of geometries , 1968 .

[12]  S. Lane.  Categories for the Working Mathematician , 1971 .

[13]  Free objects in the category of geometries , 2001 .

[14]  Thomas Zaslavsky,et al.  Biased graphs. II. The three matroids , 1991, J. Comb. Theory, Ser. B.

[15]  Nancy Ann Neudauer,et al.  Matroids You Have Known , 2009 .

[16]  D. Zangurashvili.  Several constructions for factorization systems. , 2004 .

[17]  LAURA ANDERSON,et al.  Matroid Bundles , 1998 .

[18]  Talal Ali Al-Hawary,et al.  Toward an Elementary Axiomatic Theory of the Category of LP-Matroids , 2003, Appl. Categorical Struct..

[19]  R. Diestel,et al.  Axioms for infinite matroids , 2010, 1003.3919.

[20]  Samson Abramsky,et al.  Abstract Scalars, Loops, and Free Traced and Strongly Compact Closed Categories , 2005, CALCO.

[21]  Saunders MacLane,et al.  Some Interpretations of Abstract Linear Dependence in Terms of Projective Geometry , 1936 .

[22]  S. Maclane,et al.  Categories for the Working Mathematician , 1971 .

[23]  Joseph P.S. Kung.  TWELVE VIEWS OF MATROID THEORY , 2001 .

[24]  András Recski.  Elementary strong maps of graphic matroids , 1987, Graphs Comb..

[25]  Daniel Kennedy,et al.  Majors of geometric strong maps , 1975, Discret. Math..

[26]  Horst Herrlich,et al.  Abstract and concrete categories , 1990 .

[27]  Joseph P. S. Kung,et al.  Bimatroids and invariants , 1978 .

[28]  N. White.  Theory of Matroids , 2008 .

[29]  Chris Heunen,et al.  Semimodule Enrichment , 2008, MFPS.