Lawvere theory

Template:Short description In category theory, a Lawvere theory (named after American mathematician William Lawvere) is a category that can be considered a categorical counterpart of the notion of an equational theory. Intuitively, it is a categorical generalization of algebraic structures (e.g., a group or a ring), where there exists a "generic" object and all objects are isomorphic to an integer power of $x$, representing the inputs for the $n$-ary operations on $x$ (i.e., of the form $x^n\mapsto x$, starting from the fact that $x:=x^1$ and so $x\circ x:=x^1\circ x^1:=x^1{x}^1:=x^2$; trivially generalizing inductively, we get the rest of the objects) where the operations come from the algebraic structure at hand (e.g., addition and/or multiplication).

The Lawvere theory of groups has as its generic object an underlying placeholder $x$ where the other objects are the inputs for $n$-ary operations from those integer powers of $x$ of the form ($x^n$) back to $x$, where a model, a finite-product preserving functor, from this theory into a target category $C$ such as the category of sets or topological spaces, would map the abstract theory $x$ onto the category to create a concrete, "combined," group-based structure. This model would provide a set with group structure (a group) or a topological space with group structure (a topological group), supplying appropriate names to the generic object $x$ and its mappings ($n$-ary operations) according to whatever the theory and model at play are; in the model of sets for the Lawvere theory of groups, the generic object is a group and its mappings are group operations.

Definition

Let ${\mathrm{\aleph }}_0$ be a skeleton of the category FinSet of finite sets and functions. Formally, a Lawvere theory consists of a small category $L$ with (strictly associative) finite products and a strict identity-on-objects functor $I:{\mathrm{\aleph }}_0^{\text{op}}\to L$ preserving finite products.

A model of a Lawvere theory in a category $C$ with finite products is a finite-product preserving functor $M:L\to C$. A morphism of models $h:M\to N$ where $M$ and $N$ are models of $L$ is a natural transformation of functors.

Model examples

Some examples of models of the Lawvere theory of groups (i.e., $L$ = $𝐋𝐚𝐰𝐆𝐫𝐩$):

• $M:L\to 𝐒𝐞𝐭$ ($M$ is a classical group)
• $M:L\to 𝐓𝐨𝐩$ ($M$ is a topological group)
• $M:L\to 𝐌𝐚𝐧$ ($M$ is a Lie group).

Some examples of models of the Lawvere theory of rings (i.e., $L$ = $𝐋𝐚𝐰𝐑𝐢𝐧𝐠$):

• $M:L\to 𝐒𝐞𝐭$ ($M$ is a classical ring)
• $M:L\to 𝐓𝐨𝐩$ ($M$ is a topological ring)
• $M:L\to 𝐌𝐚𝐧$ ($M$ is a smooth ring).

Category of Lawvere theories

A map between Lawvere theories $(L,I)$ and $(L^{\prime },I^{\prime })$ is a finite-product preserving functor that commutes with $I$ and $I^{\prime }$. Such a map is commonly seen as an interpretation of $(L,I)$ in $(L^{\prime },I^{\prime })$.

More formally, a map $K:(L,I)\to (L^{\prime },I^{\prime })$, such that $K\circ I=I^{\prime }$, where $I(n)=x_L^n$ and $(K\circ I)(n)=I^{\prime }(n)=x_{L^{\prime }}^n$.

Lawvere theories together with maps between them form the category $𝐋𝐚𝐰$.

Variations

Variations include multisorted (or multityped) Lawvere theory, infinitary Lawvere theory, and finite-product theory.

See also

• Algebraic theory
• Clone (algebra)
• Monad (category theory)

References

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Further reading

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