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Stream: learning: questions

Topic: Preservation of Limits

Reed Mullanix (May 17 2020 at 23:07):

In the nLab, A functor $F : C \to D$ is said to preserve a limit $x$ of a diagram $J : I \to C$ if $F(x)$ is a limit of $F \circ J$. Howerver, I've seen other sources that define preservation of limits as an isomorphim $F(lim J) \simeq lim(F \circ J)$, especially in the context of continuous functors. The 1st definition obviously induces the isomorphism, but I can't seem to figure out exactly when the isomorphism implies the 1st definition. Does anyone have any pointers?

Sources:
https://ncatlab.org/nlab/show/preserved+limit
https://ncatlab.org/nlab/show/continuous+functor

EDIT:
After playing around with this a bit, I realized that the isomorphism had to be an isomorphism of _cones_, not an isomorphism between apexes!

Soichiro Fujii (May 18 2020 at 01:24):

Precisely, a limit is not just an object, but it is defined as a (universal) cone.

So a functor $F\colon \mathcal{C}\longrightarrow \mathcal{D}$ preserves a limit $(x,p)$ of a diagram $J\colon \mathcal{I}\longrightarrow \mathcal{C}$ (where $x$ is an object of $\mathcal{C}$ and $p=(p_i\colon x\longrightarrow Ji)_{i\in\mathcal{I}}$ is the edges of a cone) if and only if $(Fx, Fp=(Fp_i)_{i\in\mathcal{I}})$ is a limit of $F\circ J$.

This is equivalent to (the existence of a limit $(y,q)$ of $F\circ J$ in $\mathcal{D}$ and) the existence of an isomorphism $(Fx,Fp)\cong (y,q)$ of cones because, in general, a cone is a limit cone if and only if it is isomorphic to some limit cone.

Nathanael Arkor (May 18 2020 at 13:24):

In response to the original question, which may have turned out not to be what you meant to ask: the paper Limit Preservation from Naturality by Caccamo–Winskel describes sufficient conditions for such an isomorphism to imply limit-preservation.

Reid Barton (May 18 2020 at 15:54):

I think this maybe obscures the main point though; if $C$ and $D$ have limits, then for any functor $F : C \to D$, there's always a canonical map $F(\lim J) \to \lim (F \circ J)$ (exercise); $F$ preserves the limit of $J$ if and only if this map is an isomorphism.

Reid Barton (May 18 2020 at 16:01):

It's somewhat tiresome to write out any of the exact definitions of "preserves limits", especially if lacking the vocabulary of limit cones, and so it's common to see abuses like "$F(\lim J) \cong \lim (F \circ J)$" to mean that the canonical map is an isomorphism or "$F(x)$ is a limit of $F \circ J$" to mean via the maps induced by $F$ and the cone morphisms for the original limit $x$ and $J$.

Nathanael Arkor (May 18 2020 at 16:15):

Right, I think this convention that $\cong$ is often a canonical isomorphism, rather than an arbitrary one, can be easy to not notice when learning, especially as this is sometimes completely implicit. (The abstract from the paper above makes this clear, though.)

John Baez (May 18 2020 at 21:39):

It's fun to try to dream up examples where $F(\mathrm{lim} J) \cong \mathrm{lim} (F \circ J)$, the mere existence of some isomorphism, holds but is not good enough.

Philip Saville (May 19 2020 at 07:46):

on the flip side, Steve Lack has a paper in which he shows that -- contrary to what you'd expect -- in certain special cases some isomorphism is actually good enough https://arxiv.org/pdf/0912.2126.pdf.