I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. Is there any reference that treat systematically the relation between such models of theories, where model means a presentation of theory?



Relation between monads, operads and algebraic theories
All nice recommendations; some more


### Lawvere theories and monads


For the connection between monads and Lawvere theories, I've found this a really nice exposition of the $\mathbf{Set}$ case:



> 
> Martin Hyland, John Power - *The category theoretic understanding of universal algebra: Lawvere theories and monads* - Electronic Notes in Theoretical Computer Science [(pdf at M Hyland's website)](http://www.dpmms.cam.ac.uk/~martin/Research/Publications/2007/hp07.pdf)
> 
> 
> 


and if what you want is something like the ultimate monads <-> Lawvere theories correspondence:



> 
> Clemens Berger, Paul-André Melliès, Mark Weber - *Monads with arities and their associated theories* [(arXiv)](http://arxiv.org/abs/1101.3064)
> 
> 
> 


Basically all of the *(Lawvere and the like) theories <-> (special) monads* equivalences can be seen as special cases of their general *monad with arities <-> theories with arities* equivalence. This paper is a real joy to read, and how all sort of nerve theorems can be viewed as an instance of the equiv between a monad with arities and the corresponding (generalized) algebraic theory is just wonderful!


Also, on the operadic side, their notion of *homogeneous* theory is related with $T$-operads (with $T$ cartesian *and* local right adjoint), and yields a nice account of *symmetric* operads as homogeneous theories with arities the Segal category $\Gamma$.


