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?
ct.category-theory – Relationship Between Monads, Operads, and Algebraic Theories
ct.category-theorymonadsoperadsreference-requestuniversal-algebra
Related Solutions
To expand on one of the points in David's answer, the absolutely crucial property of filtered colimits is that
Finite limits commute with filtered colimits in Set.
It's probably more important to know this than to know the definition of filtered colimit. In fact, you can use it as a definition, in the following sense:
Theorem Let $J$ be a small category. Then the following are equivalent:
- $J$ is filtered
- colimits over $J$ commute with finite limits in Set.
One weak point of the wikipedia article is that it gives the very concrete definition of filtered category, but it doesn't mention the following more natural-seeming formulation: a category $J$ is filtered if and only if every finite diagram in $J$ admits a cocone.
(A finite diagram in $J$ is a functor $D: K \to J$ where $K$ is a finite category. A cocone on $D$ is an object $j$ of $J$ together with a natural transformation from $D$ to the constant functor on $j$. The three conditions stated in the Wikipedia article correspond to three particular values of $K$.)
If the last couple of paragraphs have helped you, you can balance your karma by incorporating them into the Wikipedia page :-)
Lawvere theories can be thought of as "cartesian operads." That is, we have an analogy
$$\text{Lawvere theories} : \text{cartesian monoidal categories} :: \text{operads} : \text{symmetric monoidal categories}.$$
Consider the 2-category of symmetric monoidal cocomplete categories (where the monoidal structure distributes over colimits in both variables), or SMCCs for short, which you can think of as a flavor of categorified commutative rings (where colimits categorify addition and the symmetric monoidal structure categorifies multiplication). Examples include quasicoherent sheaves on some scheme or stack, in particular modules over a commutative ring.
The free such thing on a point is the symmetric monoidal category $S = \text{Psh}(\text{FinSet}^{\times})$ of combinatorial species equipped with the symmetric monoidal structure given by Day convolution starting from disjoint union, which you can think of as a categorified polynomial / power series ring. (Exercise: show that $\text{FinSet}^{\times}$ with disjoint union is the free symmetric monoidal category on a point.)
Now, if $C$ is any other SMCC, we have an equivalence $C \cong [S, C]$ of categories (where $[-, -]$ denotes homs in this 2-category: symmetric monoidal cocontinuous functors). It follows that every SMCC admits a canonical action by the monoid $[S, S] \cong S$ under composition, which is again combinatorial species, but now equipped with a new (no longer symmetric) monoidal structure, the sustitution product, which categorifies composition of power series.
Observation: An operad is precisely a monoid in $S$ with respect to the substitution product.
This is a nice exercise. The significance of this observation is that the action of $S$ on every SMCC means that monoids in $S$ correspond to natural families of monads acting on every SMCC: explicitly, if $O_n$ is an operad, the corresponding family of monads acting on every SMCC has underlying functor the corresponding "power series"
$$X \mapsto \bigsqcup_n O_n \times_{S_n} X^{\otimes n}.$$
Hence:
Slogan: Operads are natural families of monads on SMCCs.
Now you can tell exactly the same story with "monoidal" instead of "symmetric monoidal," and you will get not-necessarily-symmetric operads. You can also tell exactly the same story with "cartesian monoidal" instead of "symmetric monoidal," and you will get Lawvere theories. In more detail:
Now consider the 2-category of cartesian monoidal cocomplete categories (where products distribute over colimits in both variables), or CMCCs for short. Examples include any Grothendieck topos. The free such thing on a point is the cartesian monoidal category $F = \text{Psh}(\text{FinSet}^{op})$, which for the same reasons as above also has a substitution product. (Exercise #1: show that $\text{FinSet}^{op}$ is the free cartesian monoidal category on a point.) (Exercise #2: show that $\text{Psh}(\text{FinSet}^{op}) \cong [\text{FinSet}, \text{Set}]$, with the substitution product, is monoidally equivalent to the monoidal category of finitary endofunctors of $\text{Set}$, with the composition product.)
Observation: A Lawvere theory is precisely a monoid in $F$ with respect to the substitution product.
As above, $F$ acts on every CMCC, so Lawvere theories correspond to natural families of monads acting on every CMCC: explicitly, if $L_n$ is a Lawvere theory, the corresponding family of monads has underlying functor
$$X \mapsto \int^n L_n \times X^n$$
where the coend is indexed over the category of finite sets. (The above formula for operads is also a coend.) Hence:
Slogan: Lawvere theories are natural families of monads on CMCCs.
You might object a little to this story because you can talk about algebras over operads and models of Lawvere theories without requiring the existence of colimits, just a symmetric monoidal or cartesian monoidal structure respectively. But this isn't much of an issue in practice: just pass to the presheaf category.
Todd Trimble has written about this circle of ideas on the nLab, for example here.
This story admits very clean generalizations. For example you can try talking about braided monoidal cocomplete categories and you will get a notion of "braided operad"; you can enrich everything in sight; generally you can talk about natural families of endofunctors on any kind of category with extra structure (these correspond to endomorphisms of the forgetful functor from categories-with-extra-structure to categories), and from there you can talk about natural families of monads.
Best Answer
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:
and if what you want is something like the ultimate monads <-> Lawvere theories correspondence:
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 $\Gamma$.