### After a first course

Assuming you've completed algebraic topology from something like Hatcher's*Algebraic Topology*, Bredon's

*Geometry and Topology*or May's

*A Concise Course*

** Mosher and Tangora**

** Milnor and Stasheff**

K-theory

### Spectra

A good place to start is with Adams'*Lectures on Homology*, which is a technical master grappling with spectra.

** EKMM Spectra**

**Diagram Spectra**

Mandell, M A, J P May, S Schwede, and B Shipley. 2001. “Model Categories of Diagram Spectra.” Proceedings of the London Mathematical Society 82 (02): 441–512.

### Model Categories

A good introduction isDwyer and Spalinski

A nice more advanced source is Hirschorn's text. Ponto and May has some excellent material on model categories.

Riehl's book on Categorical Homotopy Theory is also fantastic Finally, appendix of A of Higher Topos theory by Lurie is quite readable.

### Abstract Homotopy Theory

**Vogt - Homotopy Colimits**

Vogt, R M. 1973. “Homotopy Limits and Colimits.” Mathematische Zeitschrift 134 (1): 11–52.
This paper basically proves everything you want about homotopy colimits, and it does it cleanly. Even provides a proof of one of Thomason's results

** Dwyer-Kan Simplicial Localization**

Dwyer, W G, and D M Kan. 1980. “Simplicial Localizations of Categories.” J. Pure Appl. Algebra 17 (3): 267–284.

Dwyer, W G, and D M Kan. 1980. “Calculating Simplicial Localizations.” J. Pure Appl. Algebra 18 (1): 17–35.

Dwyer, W G, and D M Kan. 1980. “Function Complexes in Homotopical Algebra.” Topology 19 (4): 427–440.

### K-theory Foundations

The first things one should probably read are

Quillen, D. 1973. “Higher Algebraic K-Theory: I.” Higher K-Theories: 85–147.

and

Waldhausen, F. 1985. “Algebraic K-Theory of Spaces.” Algebraic & Geometric Topology LMN 1126: 318–419

Both of these are papers that I have never stopped learning from - they are an endless source of both insight and tricks.