Rational homotopy theory and differential graded category

Homotopy Coherent Category Theory

Disconnected Rational Homotopy Theory

We construct two algebraic versions of homotopy theory of rational disconnected topological spaces, one based on differential graded commutative associative algebras and the other one on complete differential graded Lie algebras. As an application of the developed technology we obtain results on the structure of Maurer-Cartan spaces of complete differential graded Lie algebras.

Rational homotopy theory

1 The Sullivan model 1.1 Rational homotopy theory of spaces We will restrict our attention to simply-connected spaces. Much of this goes through with nilpotent spaces, but this will keep things easier and less technical. Definition 1. A 1-connected space X is said to be rational if either of the following equivalent conditions holds: 1. π∗X forms a graded Q-vector space. 2. H̃∗X forms a graded Q...

Coalgebraic Approach to the Loday Infinity Category, Stem Differential for 2n-ary Graded and Homotopy Algebras

We define a graded twisted-coassociative coproduct on the tensor algebra TW of any Zn-graded vector space W . If W is the desuspension space ↓V of a graded vector space V , the coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1to-1 with sequences πs, s ≥ 1, of s-linear maps on V (resp. Zn-graded Loday structures on V , sequences tha...

M392c Notes: Rational Homotopy Theory

1. Postnikov Towers and Principal Fibrations: 8/27/15 1 2. Serre Theory: 9/1/15 4 3. Rational Homotopy Groups of Spheres: 9/3/15 7 4. Commutative Differential Graded Q-algebras and Model Categories: 9/8/15 9 5. Homotopy in Model Categories: 9/15/15 13 6. Today is the Cofibrantly Generated Model Categories Day: 9/17/15 16 7. Homotopy Theory of CDGAs: 9/22/15 19 8. Minimal Sullivan Models and Sim...