Algebraic theories, span diagrams and commutative monoids in homotopy theory

Details Algebraic theories, span diagrams and commutative monoids in homotopy theory Algebraic theories, span diagrams and commutative monoids in homotopy theory

2011-09-07

arxiv.org/abs/1109.1598v1

Mathematics

Algebraic Topology

Submitted to Journal of Topology

Scientific paper

We adapt the notion of an algebraic theory to work in the setting of quasicategories developed recently by Joyal and Lurie. We develop the general theory at some length. We study one extended example in detail: the theory of commutative monoids (which turns out to be essentially just a 2-category). This gives a straightforward, combinatorially explicit, and instructive notion of a commutative monoid. We prove that this definition is equivalent (in appropriate senses) both to the classical concept of an E-infinity monoid and to Lurie's concept of a commutative algebra object.

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