Re: Weighted limits

Re: Weighted limits

[David Roberts]

Dear Jean, (apologies for this, and any future, slow replies. The
necessities of life take up a lot of my time at the moment)

My approach below is pedestrian, but I hope clear.

Strict inverters are PIE-limits. Thus they can be computed in Cat(S)
once we know it has each of products, inserters, equifers -- in fact
just the latter two, in a rather straightforward way, using no more
than two of each. To quote the nLab, "first we insert a 2-morphism
b going in the opposite direction from a, then we equify ba and ab
with identities." (this quote may be likewise borrowed from either
Kelly or Street)

Let as assume S has finite limits throughout.  For what it's worth,
products obviously exist in Cat(S).

Note that For X a category in S, and a subobject U >--> Obj(X), we can
build the full subcategory X[U] of X on U (as an object of Cat(S))
using only finite limits in S. To build the equifer of a,b: f => g: X
--> Y, we only need the equaliser E in S of the component maps a,b:
Obj(X) --> Arr(Y), and then the equifer is the inclusion  X[E] --> X
of the full subcategory on the subobject E.

Thus we are reduced to building inserters, which is the real meat of
the problem, as inserters are not equivalent to any conical 2-limit.

Consider a diagram f,g: X --> Y in Cat(S). The inserter of this
diagram is (the inclusion of) a subcategory Ins(f,g) of X. We can
compute the object Ins(f,g)_0 of objects of the inserter as the
pullback of

Obj(X) -- (f,g) --> Obj(Y) x Obj(Y) <---- (s,t) ---- Arr(Y)

in S.

Then the inserter is a wide subcategory of X[ Ins(f,g)_0 ] (itself a
full subcategory of X). Note that there is a map a: Ins(f,g)_0 --->
Arr(Y) which will be the component map of the universal natural
transformation we are inserting.

The arrows of Ins(f,g) are the largest subobject Ins(f,g)_1 --> Arr(X)
such that a is natural with respect to such arrows. This can be
defined by an equaliser in S.

Thus we can construct, using solely finite limits in S, (products,)
equifiers and inserters, and hence inverters, in Cat(S).

One could perhaps examine this proof more closely to see what kind of
internal categories in non-finitely-complete S are necessary for it to
work (eg those such that (s,t) belong to a class of which all
pullbacks exist, and are again in the class etc). This perhaps would
fit with your general philosophy on generalising fibration technology.

I hope this answers your qualms, and apologies for being slightly
telegraphic in my description.

Best regards,

David

PS I regret we did not have the chance to meet at Topos à l'IHÉS in
2015. Perhaps one day...






On 15 February 2017 at 20:09, Jean Benabou <jean.benabou@wanadoo.fr> wrote:
> Dear John,
>
> Thank you for your mail and the precisions you give in it, but I'm not
> interested, for the time being, in general questions about 2-categories.
> Let me repeat precisely my question:  If  S is a category with finite limits
> and Cat(S) is the 2-category of internal categories of S, under which
> condition does Cat(S) have strict inverters?
> Can you, or anybody give a precise answer? (Of course I know that Cat(S) is
> cotensored with 2)
> .
> David Roberts says that finite limits in S suffice. As I I said I don't
> believe that. I'm perhaps wrong. In that case, could he, you, or anybody
> tell me how to construct strict inverters when all I assume is that S has
> finite limits?
>
> It is always a pleasure to hear from you. All the best ,
>
> Jean
>
>
> Le 15 févr. 17 à 08:41, John Power a écrit :
>
> Dear Jean,
>
> Max wrote an expository paper which I believe was called "Elementary
> Observations on 2-Categorical Limits" and was published in the Bulletin of
> the Australian Mathematical Society I think around 1990. He would have had a
> discussion of inverters there.
>
> Strict inverters are a kind of strict weighted limit (see, for instance,
> https://golem.ph.utexas.edu/category/2014/04/elementary_observations_on_2ca.html)
> and a 2-category has all strict weighted limits if it has all strict conical
> limits and all strict cotensors, as a 2-category is a Cat-enriched category.
> So if one can prove that Cat(S) has strict conical limits and strict
> cotensors, one can construct strict inverters by following the procedure in
> the  link above.
>
> For strict cotensors, it suffices to prove that a 2-category has strict
> cotensors with the arrow category. I believe that is straightforward for
> Cat(S) if you follow the case of S = Set.
>
> Once again, it is always lovely to hear from you.
>
> All the best,
>
> John.
>

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Re: Weighted limits

[Richard Garner]

Dear Jean,

Here is a response which combines Ross' elegant approach with David's
elementary one.

First we construct the object of isomorphisms of any B in Cat(S). Take
the pullback of the cospan
(s,t): B_1 --> B_0 x B_0 <-- B_1: (t,s)
to get a span
p: B_1 <-- P --> B_2: q

There are induced maps (p,q):P-->B_2 and (q,p):P-->B_2 into the object
of composable pairs. Let a, b: P--> B_1 be their composites with the
composition map B_2-->B_1, and let j: Iso(B) >--> P be the pullback of
(a,b):P-->B_1 x B_1 along (i,i):B_0 x B_0 >--> B_1 x B_1 (where i is the
identities map). Iso(B) represents isomorphisms in B. There is a monic
pj : Iso(B) >--> B_1 which "forgets invertibility".

Now given a 2-cell alpha: F==>G: A-->B in Cat(S), we form its inverter
thusly. The datum for alpha is a morphism A_0-->B_1 in S. Pull back pj:
Iso(B) >--> B_1 along this to get a monic k_0: I_0 >--> A_0. The
inverter k: I >--> A of alpha has underlying graph morphism obtained by
pulling back A_1 --> A_0 x A_0 along (k_0,k_0): I_0 x I_0 >--> A_0 x
A_0.

Here is an alternative approach (which one could argue is actually the
same approach). First, for categories internal to Set, the construction
of inverters is equational and so can be done using only finite limits.
Next, if C is locally small and finitely complete, then its small
cocompletion PC also has finite limits, computed pointwise; whence
inverters for transformations between PC-internal categories  can also
be computed using only finite limits. Finally, since the Yoneda
embedding C --> PC preserves and reflects finite limits, the inverter of
a transformation between C-internal categories can also be computed
using only finite limits: one embeds into PC-internal categories,
computes the inverter there, and then notes that the finite limits used
in the computation are represented by finite limits back down in C.

Richard



On Thu, Feb 16, 2017, at 01:24 PM, Ross Street wrote:
> Dear Jean
>
> On 15 Feb 2017, at 4:47 PM, Jean Benabou
> <jean.benabou@wanadoo.fr<mailto:jean.benabou@wanadoo.fr>> wrote:
>
> However they both say that if  S is a category with finite limits the
> 2-category  Cat(S) is what I called strictly finitely 2-complete. I doubt
> it.
> Could they, or anybody else, tell me how yo prove Cat(S) has strict
> inverters?
>
> The type of limit of which you speak for 2-categories are those of
> ordinary  enriched category theory with base Cat.
>
> First agree that Cat(S) has terminal object, pullbacks and cotensoring
> with  the arrow category Arr (usually called blackboard bold 2).
>
> Next construct the category Iso with two objects and an isomorphism
> between  them using finite colimits in Cat:
> take the pushout of the two functors 2 --> Arr which are bijective on
> objects to obtain the category with two
> objects and an arrow each way; then force the arrows to be inverse to
> each other using two coequalizers.
>
> This tells us how to obtain the cotensor {Iso, A} of Iso with any A in
> Cat(S) by a pullback and two equalizers.
>
> Now take a 2-cell t between two morphisms f, g : A --> B in Cat(S). It
> corresponds to an morphism t' : A --> {Arr, B}
> in Cat(S). The inverter of the 2-cell t is the pullback of the
> restriction morphism {Iso, B} --> {Arr, B}
> along t' : A --> {Arr, B}.
>
> These things go back to my paper
>
> 10. Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra 8
> (1976) 149--181; MR53#5695.
>
> (Now I prefer the term weighted limit to indexed limit.) For any nice
> base for enrichment, all weighted limits
> can be obtained from products, equalizers and cotensoring. In the case of
> Cat, cotensoring with Arr suffices.
>
> Best wishes,
> Ross
>

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Re: Weighted limits

[David Roberts]

Dear Jean

[apologies to the moderator for sending the below message from the
wrong email address]

I must correct myself: in the paragraph

>Consider a diagram f,g: X --> Y in Cat(S). The inserter of this
>diagram is (the inclusion of) a subcategory Ins(f,g) of X. We can
>compute the object Ins(f,g)_0 of objects of the inserter as the
>pullback of
>
>Obj(X) -- (f,g) --> Obj(Y) x Obj(Y) <---- (s,t) ---- Arr(Y)
>
>in S.

I should not have said '(the inclusion of) a subcategory Ins(f,g)',
but rather 'a faithful functor Ins(f,g) --> X'.
With this change everything proceeds as before.

I can even supply a different, and cleaner, direct construction of the
inverter of the natural transformation a: f => g: X --> Y.

First define the object B of S as the pullback of

Obj(X) -- (g,f) --> Obj(Y) x Obj(Y) <---- (s,t) ---- Arr(Y)  (note
order of f and g)

as before. This gives us the projection map b': B --> Arr(Y) as noted
above, which will eventually give the putative inverse of a, and we
also have the composite map B --> Obj(X) ---a--> Arr(Y), which I will
call a_B. We can define two maps

(1)    (a_B)b': B ---(a_B,b')--> Arr(Y) x_Obj(Y) Arr(Y) --> Arr(Y)
(2)    b'(a_B): B ---(b',a_B)--> Arr(Y) x_Obj(Y) Arr(Y) --> Arr(Y)
(latter arrow is composition in both cases)

(1) gives the component of what will be a natural transformation from
f to itself, and (2) likewise, except from g to itself

Now take the equaliser of (1) and the map

B ---> Obj(X) --f--> Obj(Y) --> Arr(Y)

to get the subobject B_f --> B, and take the equaliser of (2) and the map

B ---> Obj(X) --g--> Obj(Y) --> Arr(Y)

to get the subobject B_g --> B. Now take the pullback of B_f --> B <--
B_g to get the subobject Inv(f,g)_0 ---> B. Now consider the composite
map Inv(f,g)_0 ---> Obj(X): this will be the object component of the
map from the inverter to X. Form the category J = X[ Inv(f,g)_0 ],
which has as objects Inv(f,g)_0 and as arrows the pullback (Inv(f,g)_0
x Inv(f,g)_0) x_{Obj(X) x Obj(X)} Arr(X), and comes equipped with a
fully faithful functor (in the internal sense) J --> X. Let b:
Inv(f,g)_0 ---> B --b'-> Arr(Y) be the obvious composite.

Now we need to build a wide subcategory Inv(f,g) of J and this will be
the inverter, via the given map to X. We have the component map b:
Obj(J) = Inv(f,g)_0 --> Arr(Y), but it is not necessarily natural with
respect to all the arrows of J (considered as eg generalised elements,
or in the internal language). So we consider the subobject Inv(f,g)_1
--> Arr(J), defined equationally (hence by a certain equaliser), so
that naturality squares for b commute, for arrows in Inv(f,g)_1.

Then Inv(f,g) --> J --> X is the equaliser you are looking for, and I
only used finite limits in S.

Apologies for being so long-winded, but you gave us a nice exercise
and I wanted to see it through (modulo the very last bit, I hope it is
obvious)

Best regards,
David

PS one can build all cotensors (=powers) in Cat(S) with all finite
categories using the same pedestrian logic; hence with all conical
strict 2-limits and cotensors one gets all strict weighted limits.


On 15 February 2017 at 21:03, David Roberts <a1078662@adelaide.edu.au> wrote:
> Dear Jean, (apologies for this, and any future, slow replies. The
> necessities of life take up a lot of my time at the moment)
>
> My approach below is pedestrian, but I hope clear.
>
> Strict inverters are PIE-limits. Thus they can be computed in Cat(S)
> once we know it has each of products, inserters, equifers -- in fact
> just the latter two, in a rather straightforward way, using no more
> than two of each. To quote the nLab, "first we insert a 2-morphism
> b going in the opposite direction from a, then we equify ba and ab
> with identities." (this quote may be likewise borrowed from either
> Kelly or Street)
>
> Let as assume S has finite limits throughout.  For what it's worth,
> products obviously exist in Cat(S).
>
> Note that For X a category in S, and a subobject U >--> Obj(X), we can
> build the full subcategory X[U] of X on U (as an object of Cat(S))
> using only finite limits in S. To build the equifer of a,b: f => g: X
> --> Y, we only need the equaliser E in S of the component maps a,b:
> Obj(X) --> Arr(Y), and then the equifer is the inclusion  X[E] --> X
> of the full subcategory on the subobject E.
>
> Thus we are reduced to building inserters, which is the real meat of
> the problem, as inserters are not equivalent to any conical 2-limit.
>
> Consider a diagram f,g: X --> Y in Cat(S). The inserter of this
> diagram is (the inclusion of) a subcategory Ins(f,g) of X. We can
> compute the object Ins(f,g)_0 of objects of the inserter as the
> pullback of
>
> Obj(X) -- (f,g) --> Obj(Y) x Obj(Y) <---- (s,t) ---- Arr(Y)
>
> in S.
>
> Then the inserter is a wide subcategory of X[ Ins(f,g)_0 ] (itself a
> full subcategory of X). Note that there is a map a: Ins(f,g)_0 --->
> Arr(Y) which will be the component map of the universal natural
> transformation we are inserting.
>
> The arrows of Ins(f,g) are the largest subobject Ins(f,g)_1 --> Arr(X)
> such that a is natural with respect to such arrows. This can be
> defined by an equaliser in S.
>
> Thus we can construct, using solely finite limits in S, (products,)
> equifiers and inserters, and hence inverters, in Cat(S).
>
> One could perhaps examine this proof more closely to see what kind of
> internal categories in non-finitely-complete S are necessary for it to
> work (eg those such that (s,t) belong to a class of which all
> pullbacks exist, and are again in the class etc). This perhaps would
> fit with your general philosophy on generalising fibration technology.
>
> I hope this answers your qualms, and apologies for being slightly
> telegraphic in my description.
>
> Best regards,
>
> David
>
> PS I regret we did not have the chance to meet at Topos à l'IHÉS in
> 2015. Perhaps one day...
>
>
>
>
>
>
> On 15 February 2017 at 20:09, Jean Benabou <jean.benabou@wanadoo.fr> wrote:
>> Dear John,
>>
>> Thank you for your mail and the precisions you give in it, but I'm not
>> interested, for the time being, in general questions about 2-categories.
>> Let me repeat precisely my question:  If  S is a category with finite limits
>> and Cat(S) is the 2-category of internal categories of S, under which
>> condition does Cat(S) have strict inverters?
>> Can you, or anybody give a precise answer? (Of course I know that Cat(S)  is
>> cotensored with 2)
>> .
>> David Roberts says that finite limits in S suffice. As I I said I don't
>> believe that. I'm perhaps wrong. In that case, could he, you, or anybody
>> tell me how to construct strict inverters when all I assume is that S has
>> finite limits?
>>
>> It is always a pleasure to hear from you. All the best ,
>>
>> Jean
>>
>>
>> Le 15 févr. 17 à 08:41, John Power a écrit :
>>
>> Dear Jean,
>>
>> Max wrote an expository paper which I believe was called "Elementary
>> Observations on 2-Categorical Limits" and was published in the Bulletin of
>> the Australian Mathematical Society I think around 1990. He would have had a
>> discussion of inverters there.
>>
>> Strict inverters are a kind of strict weighted limit (see, for instance,
>> https://golem.ph.utexas.edu/category/2014/04/elementary_observations_on_2ca.html)
>> and a 2-category has all strict weighted limits if it has all strict conical
>> limits and all strict cotensors, as a 2-category is a Cat-enriched category.
>> So if one can prove that Cat(S) has strict conical limits and strict
>> cotensors, one can construct strict inverters by following the procedure  in
>> the  link above.
>>
>> For strict cotensors, it suffices to prove that a 2-category has strict
>> cotensors with the arrow category. I believe that is straightforward for
>> Cat(S) if you follow the case of S = Set.
>>
>> Once again, it is always lovely to hear from you.
>>
>> All the best,
>>
>> John.
>>
>> ________________________________
>> From: Jean Benabou <jean.benabou@wanadoo.fr>
>> Sent: 15 February 2017 5:47 AM
>> To: David Roberts; John Power; Ross Street; Categories
>> Subject: Re: categories: Weighted limits

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Weighted limits

[Jean Benabou]

Dear all,

In section B1.1 of  the Elephant, weighted limits in a 2-category are
defined up to equivalence by pseudo cones. Let me say that strict
weighted limits exist if we can replace pseudo cones by genuine cones.
These limits are then defined up to a unique isomorphism.
In the category CAT of  categories (fill in universes if you don't
want to run into meta categories)  such strict limits exist.
Let me say that a 2-category is strictly finitely 2-complete if such
strict weighted limits exist for all finite (in an obvious sense)
weighted categories.

QUESTION:  Let S be a category with finite limits. I denote by Cat(S)
the 2-category of internal categories of S. Under which conditions on
S is Cat(S) strictly finitely 2-complete?

Best to all,

Jean


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