Triadic Equivalence of Regular Lawvere Theory

Regular Lawvere theories, introduced by S. Szawiel and M. Zawadowski, provide a categorical account of regular varieties in universal algebra [1]. They show that the categories of regular Lawvere theories RegLT, semi-analytic monads SanMnd, and regular operads RegOp are equivalent. In this talk, we investigate two directions in which this triadic equivalence can be generalized.

Our first generalization is to the $𝒱$-enriched setting, where $𝒱$ is a locally presentable cosmos in the sense of B’enabou. We introduce regular Lawvere $𝒱$-theories $𝒱\text{-RegLT}$ and semi-analytic $𝒱$-monads $𝒱\text{-SanMnd}$, and establish equivalence between them following [3]. We then define regular $𝒱$-operads $𝒱\text{-RegOp}$ and prove, using [2], that $𝒱\text{-RegOp}$ is equivalent to $𝒱\text{-SanMnd}$. Altogether, we obtain

$$𝒱\text{-RegLT}\simeq 𝒱\text{-SanMnd}\simeq 𝒱\text{-RegOp}.$$

As a corollary, taking $𝒱$ to be self-enriched yields a one-to-one correspondence between symmetric monoidal monads $𝖲𝒱$ on $𝒱$ and $𝒱$-enriched monads $𝖬𝒱$ on $𝒱$ [7]. Combining this with the equivalence between commutative strong monads and symmetric monoidal monads, we deduce that ${𝖲}_{𝒱}$ corresponds to a commutative regular Lawvere $𝒱$-theory, and equivalently to a regular $𝒱$-operad equipped with the Boardman–Vogt tensor product [5].

Our second generalization replaces Lawvere theories by PROPs, motivated by the fact that a Lawvere theory can be viewed as a cartesian PROP. We define a notion of regular PROP rProp and prove its equivalence with regular colored operads cRegOp via the adjunction studied in [6]. Moreover, we show that rProp is equivalent to the category of semi-analytic monads on polygraphs ${\text{SanMnd}}_{𝐩}$ [4]. Consequently, we obtain

$$\text{rProp}\simeq \text{cRegOp}\simeq {\text{SanMnd}}_{𝐩}$$

.