Re: Weighted limits

[Richard Garner <richard.garner@mq.edu.au>]

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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