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Stream: theory: category theory

Topic: Computing pseudo-limits and pseudo-colimits

Jens Hemelaer (Dec 09 2020 at 13:32):

Suppose you have a strict 2-category $\mathcal{C}$, a small 1-category $I$, and a functor $I \to \mathcal{C}$ given by $i \mapsto X_i$.
Take $X$ with maps $X \to X_i$ such that the resulting diagrams commute up to isomorphism.

Is it true that $X$ is a 2-limit / pseudo-limit if and only if the functor
$\mathrm{Hom}(C,X) \to \mathrm{lim}\, \mathrm{Hom}(C,X_i)$
is an equivalence of categories for all $C$? The limit on the right hand side is a pseudo-limit in the 2-category of small categories.
Does it still work if you compute the pseudo-limit on the right hand side in the (2,1)-category of small categories (i.e. with only natural isos and no general natural transformations)? What if you compute it in the $(\infty,1)$-category of small categories?

Similarly, for maps $X_i \to X$ in the other direction, is $X$ a 2-colimit / pseudo-colimit if and only if
$\mathrm{Hom}(X, C) \to \mathrm{lim}\,\mathrm{Hom}(X_i,C)$
is an equivalence of categories for all $C$?

Reid Barton (Dec 09 2020 at 13:54):

I assume by "commute up to isomorphism" in the second sentence you mean commute up to coherent isomorphism.

Reid Barton (Dec 09 2020 at 13:55):

By "pseudolimit", I usually understand a specific model for the limit up to isomorphism, not equivalence. Then I would answer yes to the first question if "equivalence" is replaced by "isomorphism". For 2-limit more generally, I would answer yes as stated.

Reid Barton (Dec 09 2020 at 13:55):

I'm not sure if this understanding is universal though.

Jens Hemelaer (Dec 09 2020 at 14:01):

Thanks! I forgot the coherence part. What do you mean by universal?

Reid Barton (Dec 09 2020 at 14:02):

... as in agreed to by all higher category theorists. :upside_down:

Reid Barton (Dec 09 2020 at 14:05):

I thought I got this convention from the nLab but now I don't see it at the moment.

Jens Hemelaer (Dec 09 2020 at 14:05):

That may be why I didn't see it in print somewhere...

So, what is the reason that people disagree? Is there a proof that people disagree over or maybe the definition of 2-limit?

Reid Barton (Dec 09 2020 at 14:06):

Ah, I found https://ncatlab.org/nlab/show/2-limit#Terminology. They use the phrase "strict pseudolimit" which is certainly clearer

Jens Hemelaer (Dec 09 2020 at 14:11):

I used the terminology pseudo-limit because "2-limit" seems to have multiple meanings.

The one I mean can then be described as "not necessarily strict pseudolimit".

Reid Barton (Dec 09 2020 at 14:19):

Someone with the viewpoint "2-category = strict 2-category = Cat-enriched category" will probably think about strict 2-limits by default, and even if they use some replacement for the weight of the limit to make it equivalence-invariant (e.g. a (strict) pseudolimit), they might still expect the limit to be determined up to isomorphism and not just equivalence.
On the other hand, someone coming from a weak/bicategorical or model-independent perspective would use a weaker, up-to-equivalence definition by default, since even the notion of "isomorphism" itself is specific to the model of strict 2-categories.

Reid Barton (Dec 09 2020 at 14:20):

So we ended up with (at least) two conflicting sets of terminology for the same notions.

Jens Hemelaer (Dec 09 2020 at 14:34):

Ah, I don't mind that the 2-limit is only determined up to equivalence, I just wanted to exclude lax limits.
Thanks for the explanation, I didn't know there were multiple ideas of what a strict 2-category is.

Two follow-up questions that are hopefully not too off-topic:
What are the weights in this (non-weighted) case?
And what properties do the functors $X_i \to X_j$ need to have such that the 2-limit is the ordinary 1-categorical limit?
I assume that they need to be cofibrations or fibrations but I don't know what kind of functors these are.
Hopefully, there is still "no weight".

Reid Barton (Dec 09 2020 at 14:38):

For the last question I don't think there is a simple answer except in special circumstances--for example, in a pullback it's enough for one of the maps to be an isofibration

Reid Barton (Dec 09 2020 at 14:39):

There's only one notion of "strict 2-category" but there are multiple ideas of what 2-category theory is about in general.

Jens Hemelaer (Dec 09 2020 at 14:42):

Reid Barton said:

For the last question I don't think there is a simple answer except in special circumstances--for example, in a pullback it's enough for one of the maps to be an isofibration

Thanks! I'm mostly interested in pullbacks anyway at the moment. The isofibration criterion looks very concrete and useful, I'll see if I can apply it in an example.