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Algebras for endofunctors bridge the gap between functors acting on a category and structures defined in it. An algebra for an endofunctor $F$ is instantiated by some morphism $Fa \to a$, and more crucially a morphism of algebras is a map $a\to b$ between their codomains making the evident diagram commute. This construction is intrinsically homotopical, so it makes good sense to loosen the restriction from just endofunctors.

Do you know of any writing about algebras for more general transfors? Seems that many 2-categorical results about algebras over monads, lax morphisms, et cetera, would enjoy a more full account of transfors and their algebras.

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    $\begingroup$ What do you mean when you say "this construction is intrinsically homoropical"? What's homotopical about it? $\endgroup$
    – Alex Kruckman
    Commented May 18, 2021 at 12:54
  • $\begingroup$ An algebra for $F$ is a "homotopy" (natural transformation) $F\bar{a} \to \bar{a}$, where $\bar{a}$ denotes any constant functor. I would go so far as to say that the current definition is not poised for a proper homotopical generalization, because constant functors are an "evil" notion internally. The emphasis must shift away from the "carriers" of algebras and toward the syntactic activity that specifies the algebra in an existing syntactic context (a relevant category). Emphasizing the homotopies themselves over the nullhomotopic (constant) maps. $\endgroup$
    – Mathemologist
    Commented May 18, 2021 at 13:50
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    $\begingroup$ The property of a given functor "being constant" (strictly) is "evil", but "the constant functor associated to a given object" as a construction is not evil. $\endgroup$
    – Mike Shulman
    Commented May 18, 2021 at 14:00

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The category of algebras for an endofunctor $F:C\to C$ is the lax limit of the 2-functor $[F]:D\to Cat$, where $D$ is the 2-category freely generated by one object and one endomorphism. Accordingly, it seems natural to me that more general lax limits could be regarded as categories of "algebras for transfors".

For instance, if instead we take $D$ to be the 2-category freely generated by one object, two endomorphisms $f,g$ of that object, and a 2-cell $f\to g$, then a diagram $D\to Cat$ consists of two endofunctors $F,G$ of a category $C$ and a transformation $\alpha:F\to G$. Then an object of the lax limit could be regarded as an "algebra for $\alpha$". Concretely, this consists of an object $X$ with maps $FX\to X$ and $GX\to X$ making a triangle commute with $\alpha_X$. Of course this is equivalent to just making $X$ a $G$-algebra, so algebras for a natural transformation aren't a new thing.

However, other lax limits give examples of other important and related structures. For instance, the category of monad-algebras for a monad, and the category of pointed-algebras for a pointed endofunctor, are both lax limits. And one can likewise write down different kinds of lax limits of higher categories to represent other algebra-like structures.

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