homotopy theory of inverse semigroups


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

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Basic facts


Homotopy theory of inverse semigroups


Methods from abstract homotopy theory can be used to define a suitable notion of homotopy equivalence for inverse semigroups. As an application of this theory, one can prove a theorem for inverse semigroup homomorphisms which is the exact counterpart of the well-known result in topology which states that every continuous function can be factorised into a homotopy equivalence followed by a fibration.

In the paper LMP it is shown that this factorisation is isomorphic to the one constructed by Steinberg in his Fibration Theorem, originally proved using a generalisation of Tilson’s derived category.


  • M. V. Lawson, J. Matthews, and T. Porter, The homotopy theory of inverse semigroups , IJAC 12 (2002) 755-790.

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