Timeline for Conceptual reason that monadic functors create limits?
Current License: CC BY-SA 4.0
7 events
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| Feb 28, 2020 at 4:24 | comment | added | Tim Campion | I can't resist mentioning that another cute application of this theorem is a proof of the Knaster-Tarski fixed point theorem. The proof I have in mind proceeds in two steps, first viewing complete lattices as inf-complete, and then viewing them as sup-complete. | |
| Feb 22, 2020 at 0:47 | comment | added | Mike Shulman | You can deduce the corresponding result for limits of a particular diagram using the Yoneda embedding, which preserves and reflects all existing limits. A monad $T$ on $C$ induces by left Kan extension a monad $\hat{T}$ on the presheaf category $\hat{C}$, and the latter is complete. Thus the category of $\hat{T}$-algebras has all limits created in $\hat{C}$, and a $\hat{T}$-algebra is a $T$-algebra precisely when its underlying object in $\hat{C}$ is representable. See Prop. 5.6 of arxiv.org/abs/1104.2111 for a similar argument. | |
| Feb 22, 2020 at 0:46 | comment | added | Mike Shulman | An especially cute thing about this is that it's an example of the microcosm principle: we show that categories of algebras for 1-monads (i.e. monads in 2-categories) create limits by showing that 2-categories of algebras for 2-monads (i.e. monads on 2-categories, i.e. monads in 3-categories) create limits. | |
| Feb 18, 2020 at 12:11 | comment | added | john | Yes, that's right. With regards (3), it does cover the case of an endofunctor too -- that's the lax limit of an endomorphism. I forgot to mention this. I agree it doesn't answer your (2). With regards (1) one can certainly capture creation in the 2-monad sense too though, much as for (2), it won't concern an "individual limit". | |
| Feb 18, 2020 at 6:59 | comment | added | Tim Campion | Of course, this is a bit weaker than than the result I asked about, in a few respects: (1) we get that the existence of certain limits in $C^T$, but not the full strength of "creating a limit" (2) we need to assume that $C$ has all $D$-limits, not just the limit of a particular diagram (3) we don't get the case of algebras for an endofunctor. I think (1) is not a big deal since one can show separately that the forgethful functor preserves limits and is conservative. (2) and (3) are more significant weakenings. Nevertheless, it's really great to get this far at such an abstract level! | |
| Feb 18, 2020 at 6:56 | comment | added | Tim Campion | Let's see if I've got this -- Let $\mathbb T: Cat \to Cat$ be the 2-monad for categories with $D$-limits. Then $\mathbb T-Alg_c$ is the 2-category of categories with $D$-limits and all functors between them. By Lack's result the forgetful functor $\mathbb T-Alg_c \to Cat$ creates lax limits. In particular, a monad $T: C \to C$ in $\mathbb T-Alg_c$ is the same as a monad in $Cat$ whose underlying category $C$ has $D$-limits. So Lack's result says that the EM object for $T$ in $\mathbb T-Alg_c$ is the usual category of algebras $C^T$ as in $Cat$. In particular, $C^T$ has $D$-limits. Neat! | |
| Feb 17, 2020 at 17:59 | history | answered | john | CC BY-SA 4.0 |