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Stream: theory: category theory

Topic: The free F-algebra is a T-algebra

Patrick Nicodemus (May 27 2023 at 19:33):

In my research I have been coming across hypotheses and definitions like this:

• $C$ and $D$ are categories, $F$ is a functor $C\to D$. $T$ is a monad on $D$. An $(F,T)$ structure on an object $X$ in $C$ is a $T$-algebra structure on $F(X)$. A morphism of $(F,T)$-structures is a map $f:X\to Y$ such that $F(f)$ is a morphism of $T$-algebras. (Maybe we could also say that $X$ is a $T$-algebra relative to $F$.)

• A special case of the above. $C$ is a category. $T$ and $F$ are both monads on $T$. There is a given distributive law $\alpha :T\Rightarrow F$ so that $T$ operates on $F$-algebras in a natural way. An $(F,T)$ structure on $X$ in $C$ is a $T$-algebra structure on the free $F$-algebra $(FX, \mu^F_X)$.

Replace monad with comonad, etc.

I would like to ask what examples of this situation you can think of and where you tend to observe them. What papers have needed similar hypotheses?

Ralph Sarkis (May 28 2023 at 00:20):

This rings some bells from the interaction of algebras and colagebras in the generalized determination literature e.g. this.

Patrick Nicodemus (May 28 2023 at 01:55):

Yes, indeed this looks very nice!

Patrick Nicodemus (May 28 2023 at 03:00):

Has anybody ever heard of a theorem of the following form:

Let $F$ be a monad on $C$ and $T$ a monad on $D$. Then both $F$ and $T$ act on the functor category $[C;D]$.

Assume $X : C\to D$ has the property that $X\circ F$ is a $T$-algebra, and moreover the $T$-algebra structure is coherent with that of $F$. Then, (under some additional hypotheses on $F,T$), the limit of $X$ is a $T$-algebra.

Patrick Nicodemus (May 28 2023 at 03:03):

It is a variant of the idea that the limit of algebras is again an algebra but somehow it is much stronger because we do not require that all objects in the diagram $X$ are algebras, or all the morphisms in $X$ algebra morphisms. Only those which are the images of free $F$-algebras and algebra maps between them. Maybe I need additional hypotheses which involve that the $F$-algebras are somehow "dense" enough in $C$.