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Stream: theory: category theory

Topic: Lifting FS to algebras

fosco (Jun 02 2020 at 11:13):

What kind of conditions can one put on a triple $(\mathcal C,T,(\mathcal E,\mathcal M))$, where $\mathcal C$ is a category, $T$ is a monad on $\mathcal C$, and $(\mathcal E,\mathcal M)$ is a factorisation system on $\mathcal C$, ensuring that there is a factorisation system $(\mathcal E^T,\mathcal M^T)$ on the category of $T$-algebras?

Reid Barton (Jun 02 2020 at 11:28):

Well, if $\mathcal{C}$ and $\mathcal{C}^T$ are locally presentable, and $\mathcal{E}$ is generated by a set of maps... is this not the kind of condition you had in mind?

fosco (Jun 02 2020 at 11:32):

Orthogonality shall follow from niceness of $\mathcal C$ and $T$, yes; but how can I be sure the factoring object is an algebra?

fosco (Jun 02 2020 at 11:33):

More specifically, I have a precise example in mind: C is Cat, $T$ is the free monoid monad; $(E,M)$ is the bo-ff factorisation: how can I be sure that the factorisation of a strong monoidal functor between monoidal categories is a monoidal category, and the bo and ff parts are strong monoidal?

Reid Barton (Jun 02 2020 at 11:54):

Oh, did you mean $\mathcal{E}^T$ is algebra maps whose underlying map belongs to $\mathcal{E}$, and similarly for $\mathcal{M}$?

fosco (Jun 02 2020 at 12:08):

yes!

Reid Barton (Jun 02 2020 at 12:17):

It looks like maybe it's necessary and sufficient that $T$ preserves the class $\mathcal{E}$

Reid Barton (Jun 02 2020 at 12:19):

Necessary because if $(\mathcal{E}^T, \mathcal{M}^T)$ is an OFS on $\mathcal{C}^T$ then the forgetful functor to $\mathcal{C}$ preserves the right class, which means the free functor $\mathcal{C} \to \mathcal{C}^T$ preserves the left class, which is to say $T$ preserves $\mathcal{E}$

Reid Barton (Jun 02 2020 at 12:21):

For sufficiency I'm not so sure but mainly we need to check that we can factor, so suppose $A \to C$ is a map in $\mathcal{C}^T$ and we factored the underlying map of $\mathcal{C}$ as $A \to B \to C$ as an $\mathcal{E}$-map followed by an $\mathcal{M}$-map. We still have to construct $TB \to B$ which fits in a diagram with $TA \to A$ and $TC \to C$.

Reid Barton (Jun 02 2020 at 12:22):

If you redraw the diagram with $TA \to TB$ on the left and $B \to C$ on the right, then $TA \to TB$ belongs to $\mathcal{E}$ (by the assumption on $T$) and $B \to C$ belongs to $\mathcal{M}$

Reid Barton (Jun 02 2020 at 12:22):

so then that gives you a map $TB \to B$, and I imagine the remaining conditions will follow from uniqueness of lifts