Abstract
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.
Highlights
Matroids are very general structures that capture the notion of independence
One feature of matroid theory that we leave for future work is duality: functoriality of this construction needs a choice of morphisms that stands to strong maps as relations stand to functions
We first focus on pointed matroids, which play an important role in matroid theory
Summary
Matroids are very general structures that capture the notion of independence. They were introduced by Whitney in 1935 as a common generalization of independence in linear algebra and independence in graph theory [26]. They were linked to projective geometry by Mac Lane shortly thereafter [18], and have found a great many applications in geometry, topology, combinatorial optimization, network theory, and coding theory since [21, 25]. Matroids have mostly been studied individually, and most authors do not consider
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