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sarahzrf (Nov 30 2020 at 19:36):i want to consider categories equipped with endofunctors, say (C, F) and (C', F'), and then consider functors p : C → C' equipped with natural isomorphisms image.png
sarahzrf (Nov 30 2020 at 19:37):then p lifts to a functor between the categories of F-algebras and F'-algebras, and between the categories of F-coalgebras and F'-coalgebras
sarahzrf (Nov 30 2020 at 19:39):a bit more systematically, we can assemble such pairs (C, F) into the objects of a 2-category, and functors equipped w/ such natural isos as the 1-cells—put shortly, it's the 2-category of [strict for convenience, if you prefer] 2-functors BN → Cat—and then i think "take the category of [co]algebras" should be a 2-functor from this to Cat (in particular, it's "take the [op]lax limit of the functor BN → Cat")
(here BN is the one-object category corresponding to the additive monoid of natural numbers, i.e., the free monoid on one generator)
sarahzrf (Nov 30 2020 at 19:40):now i'd like to know: what kind of properties does this lifting of p have, and how do they relate to the properties of p?
sarahzrf (Nov 30 2020 at 19:40):in particular, i'd like to be able to show that the lifting preserves initial algebras and/or terminal coalgebras under certain conditions
sarahzrf (Nov 30 2020 at 19:41):is there somewhere i can read about any of these constructions?
sarahzrf (Dec 03 2020 at 20:56):...hmmm, i think if
then among other things, p preserving the terminal object of C should imply that its lifting preserves the terminal coalgebra
sarahzrf (Dec 03 2020 at 20:57):i wonder if such a beck-chevalley condition tends to be satisfied?
Morgan Rogers (he/him) (Dec 03 2020 at 22:13):Always hard to answer these questions without a source of examples to hand!
sarahzrf (Dec 03 2020 at 23:03)::sweat_smile:
sarahzrf (Dec 03 2020 at 23:04):okay, the primary example i have in mind is where the categories are Top_* and Grp, p = π₁, and the endofunctors are F(X) = X ∨ S¹ and F'(G) = G + Z
sarahzrf (Dec 03 2020 at 23:06):then im pretty sure we have a natural isomorphism image.png
sarahzrf (Dec 03 2020 at 23:07):so that gives us a lift of π₁ to a functor from the category of F-algebras to the category of F'-algebras, and a lift to a functor from the category of F-coalgebras to the category of F'-coalgebras
sarahzrf (Dec 03 2020 at 23:08):i happen to know that the carriers of the terminal coalgebras of F and F' are the hawaiian earring and its fundamental group, respectively, so i am confident that the lift to the category of coalgebras will preserve the terminal coalgebra
sarahzrf (Dec 03 2020 at 23:08):i'm pretty sure the initial algebra gets preserved too
sarahzrf (Dec 03 2020 at 23:08):but i'm hoping there's a way to show this without already knowing what the group is, because then that would be a way of computing the group
Morgan Rogers (he/him) (Dec 04 2020 at 09:43):Just to check, you're considering the algebras/coalgebras for the endofunctor here, right? Rather than the monad/comonad it carries?
I ask because in this example, if my mental calculation is correct, the endofunctors carry both monad and comonad structures.
sarahzrf (Dec 04 2020 at 14:09):right
sarahzrf (Dec 04 2020 at 14:10):no [co]monad structures intended here
Joshua Meyers (Feb 10 2021 at 03:23):Just started a new topic that is relevant: https://categorytheory.zulipchat.com/#narrow/stream/229136-theory.3A-category.20theory/topic/Categories.20of.20Algebras/near/225789981