Re: comma categories

Re: comma categories

[categories]

Date: Tue, 1 Jul 1997 11:38:17 -0400
From: Michael Barr <barr@triples.math.mcgill.ca>

You are right.  It is called slice categories.

Michael Barr
>From john@cs.keele.ac.uk Tue Jul  1 04:09:41 1997
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Subject: comma categories
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> Not that I am blowing my own horn (but why not), you will find something
> about comma categories in Barr & Wells, Category Theory for Computing
> Science.  But no, I know of no other research along those lines.
> 
> Michael Barr

Pehaps it would be helpful to the original enquirer to say where
it can be found. I am unable to find it in the index of either edition.


John Stell

Re: Comma categories

[Steve Lack]

On 25/09/09 6:23 AM, "Tony Meman" <tonymeman1@googlemail.com> wrote:

> Dear category theorists,
> I have two questions concerning comma categories.
>
> If C is a category with a terminal object *, is the comma category (C,*)
> consisting of arrows from C to * isomorphic to the category C itself? If
> this is true, the same should apply to the dual case with an initial object.
>

Dear Tony,

Yes, this is true. You could even take it as a definition of terminal
object.

> The category sSet of simplicial sets is the category of functors from the
> opposite delta category Delta^op to Set. The category of pointed simplicial
> sets sSet* is defined as the comma category (delta0, sSet), where
> delta0=hom(-,[0]). Is sSet* isomorphic to the category of functors from
> ([0],Delta)^op to Set?
>

No, this is not true. The category sSet* is pointed (it has a terminal
object which is also initial), while the category of functors from
([0],Delta)^op to Set is not.

I'm not sure if there was supposed to be a connection between the two
questions, but just in case, I might point out that [0] is not initial in
Delta (in fact it is terminal).

Steve Lack.

> Thank you in advance for any help.
> Tony
>
>



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Comma categories

[Tony Meman]

Dear category theorists,
I have two questions concerning comma categories.

If C is a category with a terminal object *, is the comma category (C,*)
consisting of arrows from C to * isomorphic to the category C itself? If
this is true, the same should apply to the dual case with an initial object.

The category sSet of simplicial sets is the category of functors from the
opposite delta category Delta^op to Set. The category of pointed simplicial
sets sSet* is defined as the comma category (delta0, sSet), where
delta0=hom(-,[0]). Is sSet* isomorphic to the category of functors from
([0],Delta)^op to Set?

Thank you in advance for any help.
Tony


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Comma categories

[Robert L Knighten]

Bill Lawvere writes:
 >
 >
 > I recently noticed that in Abstract no. 652-4 in the Notices
 > of the AMS volume 14 (1967) page 937, John Gray advocates a
 > systematic treatment of the calculus of comma categories
 > and lists five operations which should be explicitly accounted
 > for in such a calculus.
 > He also mentions that Jon Beck contributed to that discussion.
 >
 > Probably John Gray's notes, if they still exist, would be
 > a helpful guide to someone planning to write a systematic
 > treatment as suggested recently Uwe Wolters.
 >
 > Bill

As a followup to Bill's note, here is a slightly more recent positing by John
Gray to another mailing list on this very topic.

    * To: types@theory.LCS.MIT.EDU
    * Subject: Re: Cobig, Coproduct, and Comma
    * From: gray@symcom.math.uiuc.edu (John Gray)
    * Date: Mon, 20 Mar 89 17:13:53 EST
    * Sender: meyer@theory.LCS.MIT.EDU

Date: Mon, 20 Mar 89 15:32:11 CST

>Cobig, Coproduct, and Comma  Vaughan Pratt  3/19/89
>Formally a comma category is most slickly described as a lax pullback.
>I've attempted an understandable account of this 2-category concept in
>an appendix below.  I'd appreciate pointers to other accounts.

Comma categories are an ancient tool in category theory.
They were introduced in
	F. W. Lawvere, Functorial Semantics of Algebraic Theories
	Thesis, Columbia University, 1963.
He used them in
	 --, The category of categories as a foundation for
	mathematics, Proceedings of the Conference on Categorical
	Algebra, La Jolla 1965, Springer-Verlag, New York.
I discussed them in several places:
	J. W. Gray,  Fibred and cofibred categories, same proceedings
	as above, 21-83.
I gave a brief calculus of comma categories in:
	--, The categorical comprehension scheme, Category theory,
	Homology theory and their Applications III, Lecture Notes in
	Mathematics 99, Springer-Verlag, New York 1969, 242-312.
They are described as "Cartesian quasi-limits" in the book:
	--, Formal category theory: Adjointness for 2-categories,
	Lecture Notes in Mathematics 391, Springer-Verlag, New York 1974.
which is the first place where the lax description of them can be found.
I don't credit it to anybody there, since I assumed it was general knowledge.
The name was changed to "lax limits"  in:
	G. M. Kelly and R. Street, Review of the elements of 2-categories,
	Category Seminar, Lecture Notes in Mathematics 420, Springer-
	Verlag, New York 1974.
The general theory of the properties of lax limits in 2-categories was
discussed independently by Street and me in various publications.  E. g.,
	J. W. Gray, The existence and construction of lax limits,
	Cahiers Top. et Geom. Diff. 21 (1980), 277-304.
	--, Closed categories, Lax limits and homotopy limits, J. Pure
	Appl. Algebra 19 (1980), 127-158.
	--, The representation of limits, lax limits, and homotopy limits
	as sections, in Mathematical Applications of Category Theory,
	Contemporary Mathematics 30 (1984), AMS, 63-83.
	R. Street, Two constructions on lax functors, Cahiers Top. et
	Geom. Diff. 13, (1972), 217-264.
	--, Limits indexed by category-valued 2-functors, J. Pure and
	Applied Alg. 8 (1976), 149-181.

It is of course very gratifying to see these ideas coming around again as
useful tools in the semantics of programming languages.

	John Gray


-- Bob

Re: Comma categories

[Bill Lawvere]

I recently noticed that in Abstract no. 652-4 in the Notices
of the AMS volume 14 (1967) page 937, John Gray advocates a
systematic treatment of the calculus of comma categories
and lists five operations which should be explicitly accounted
for in such a calculus.
He also mentions that Jon Beck contributed to that discussion.

Probably John Gray's notes, if they still exist, would be
a helpful guide to someone planning to write a systematic
treatment as suggested recently Uwe Wolters.

Bill


On Mon, 5 Nov 2007, claudio pisani wrote:

>
> The following facts about slice categories may be
> worth noticing:
>
> 1 In the equivalence between df/X (discrete fibrations
> over a category X) and presheaves on X, the slices X/x
> -> X correspond to the representable presheaves.
>
> 2. (Yoneda Lemma) The reflection of x:1->X (as an
> object of Cat/X) in df/X is (isomorphic to) X/x (with
> its terminal object as reflection map).
> In particular, the full subcategory sl/X of df/X
> generated by the slices over X is isomorphic to X.
>
> 3. For any functor p:P->X, a morphism p->X/x in Cat/X
> is a cone of base p and vertex x.
>
> 4. So, a reflection of p->X/x of p in sl/X is a
> colimiting cone.
>
> 5. A functor f:X->Y has a right adjoint iff the
> pullback f*Y/y of any slice of Y is (isomorphic to) a
> slice of X.
>
> 6. If ex_f -| f* : df/Y -> df/X is the "left Kan
> extension" along f, then the counit
> e: ex_f f* Y/y -> Y/y
> is an iso for any y iff f is "dense" (aka "connected")
> while it is a colimiting cone for any y iff f is
> "adequate" (aka "dense").
> Using instead the adjunction
> f_! -| f* : Cat/Y -> Cat/X
> the counit is a colimiting cone for any y iff f is
> adequate (as before), while it is an absolute colimit
> iff f is dense.
>
> Best regards.
>
> Claudio
>
>
>
> --- Uwe Egbert Wolter <Uwe.Wolter@ii.uib.no> ha
> scritto:
>
>> Dear all,
>>
>> I'm looking for a comprehensive exposition of
>> definitions and results
>> around comma/slice categories.  Especially, it would
>> be nice to have
>> something also for non-specialists in category
>> theory as young
>> postgraduates. Is there any book or text you would
>> recommend?
>>
>> Best regards
>>
>> Uwe Wolter
>>
>>
>>
>
>
>
>

Re: Comma categories

[claudio pisani]

The following facts about slice categories may be
worth noticing:

1 In the equivalence between df/X (discrete fibrations
over a category X) and presheaves on X, the slices X/x
-> X correspond to the representable presheaves.

2. (Yoneda Lemma) The reflection of x:1->X (as an
object of Cat/X) in df/X is (isomorphic to) X/x (with
its terminal object as reflection map).
In particular, the full subcategory sl/X of df/X
generated by the slices over X is isomorphic to X.

3. For any functor p:P->X, a morphism p->X/x in Cat/X
is a cone of base p and vertex x.

4. So, a reflection of p->X/x of p in sl/X is a
colimiting cone.

5. A functor f:X->Y has a right adjoint iff the
pullback f*Y/y of any slice of Y is (isomorphic to) a
slice of X.

6. If ex_f -| f* : df/Y -> df/X is the "left Kan
extension" along f, then the counit 
e: ex_f f* Y/y -> Y/y 
is an iso for any y iff f is "dense" (aka "connected")
while it is a colimiting cone for any y iff f is
"adequate" (aka "dense").
Using instead the adjunction 
f_! -| f* : Cat/Y -> Cat/X
the counit is a colimiting cone for any y iff f is
adequate (as before), while it is an absolute colimit
iff f is dense.

Best regards.

Claudio



--- Uwe Egbert Wolter <Uwe.Wolter@ii.uib.no> ha
scritto:

> Dear all,
> 
> I'm looking for a comprehensive exposition of
> definitions and results
> around comma/slice categories.  Especially, it would
> be nice to have
> something also for non-specialists in category
> theory as young
> postgraduates. Is there any book or text you would
> recommend?
> 
> Best regards
> 
> Uwe Wolter
> 
> 
> 

Re: Comma categories

[wlawvere]

Dear Uwe
You are right in thinking that there should be such 
an exposition because the construction is explicitly 
or implicitly involved in so many contexts that a 
formal summary would be useful. Unfortunately, 
I know of no such exposition though Hugo Volger 
started one many years ago.

As you can see from the TAC Reprint of my 
thesis, the original motivation was to 
be able to state the definition of adjointness in a 
wholly elementary way for arbitrary categories 
without involving enrichments in some fixed
category of sets. If A is a reflective subcategory in 
some X and if B is coreflective in the same X, then 
composing the implicit functors yields an adjoint 
pair between A and B. The point is that conversely 
any adjoint pair can be so factored through a third 
"adjunction" category X and the universally available 
choice has this simple construction as a pullback.
It proved to be the appropriate tool for calculating
Kan extensions, adequacy comonads, fibrations,etc.
Grothendieck defined slice categories and Artin the 
gluing, both of which are special cases of this
construction.

Although inserters are interdefinable (like equalizers
vs pullbacks), some consider inserters more basic: 
given x:A->C and y:B->C, one can take the 
inserter of the two composites AxB->C to obtain 
the construction under discussion. 

In the special case A=B=1 (when the inserter and the 
"comma" category are the same) we obtain the homset 
(x,y) of two objects of C. The latter was the reason 
for my notation: it generalizes a frequent notation for 
hom.[Recall that every object belongs to a unique 
category; thus the standard notation C(x,y) is
actually redundant (if C is not enriched), though easier
to understand. Either notation is preferable to the 
excessive HomsubC, a back formation not be confused
with the informative HomsubR when C arises from 
adjoining some additional structure R to a given base.]
 
Concerning the bizarre name:
(1) I had neglected to give the construction any name, 
so (2) one started giving it a name based on reading 
aloud the notation: x comma y; (3) some continued
the "name" but changed the notation to a vertical arrow.

Since it is well justified to name a category for its 
objects, and since the effect of insertion is to create 
objects with one ingredient more of structure, recent 
discussions here have proposed the name/notation
             Map(x,y)
[or for emphasis Map(subC)(x,y)]
for the category with its faithful functor to AxB.

Although I often use the word "map" interchangeably
with "morphism", note that the above suggests a more
concrete content: philosophically, in order to confront 
objects in two categories A and B, it is necessary to 
first functorially transport them into a common 
category C. For example to map a 2-truncated simplicial 
set to a diffentiable manifold (such as a piece of
paper) one first interprets each in appropriate ways as 
topological spaces, and the resulting objects form a 
category (having full subcategories of "cartographical" 
interest).  

I would be happy to offer a prize for the best exposition!

Bill

Quoting Uwe Egbert Wolter <Uwe.Wolter@ii.uib.no>:

> Dear all,
> 
> I'm looking for a comprehensive exposition of definitions and
> results
> around comma/slice categories.  Especially, it would be nice to have
> something also for non-specialists in category theory as young
> postgraduates. Is there any book or text you would recommend?
> 
> Best regards
> 
> Uwe Wolter
> 
> 
> 
> 

Comma categories

[Uwe Egbert Wolter]

Dear all,

I'm looking for a comprehensive exposition of definitions and results
around comma/slice categories.  Especially, it would be nice to have
something also for non-specialists in category theory as young
postgraduates. Is there any book or text you would recommend?

Best regards

Uwe Wolter

Re: Comma categories

[F W Lawvere]

A crucial point is whether the recipient of the enriching is cartesian or
not. Note that fully internalising always must involve a cartesian aspect
since one must diagonalize on the parametrizers of families of objects 
(at least) in order to explain eg natural transformations, even if the
parametrizers for individual homs are not cartesian (eg linear or metric).

One can envisage replacing individual "comma" categories by families of
categories parametrized by (commutative) coalgebras, which seems just a
way of constructing a cartesian category for the purpose, to which it may
or may not be adequate. 

Symmetric monoidal categories in which the unit object is terminal seem to
have a special role, but that may be illusory.(After all "any" smc is
covered by one with that additional property) . Perhaps the affine modules
( see my paper "Grassmann's dialectics and category theory") constitute a
good test case for proposed constuctions

Bill Lawvere.
*******************************************************************************
F. William Lawvere			Mathematics Dept. SUNY 
wlawvere@acsu.buffalo.edu               106 Diefendorf Hall
716-829-2144  ext. 117		        Buffalo, N.Y. 14214, USA

*******************************************************************************
                       


On Mon, 19 Oct 1998, Manuel Bullejos wrote:

> 
> Does any body know if comma categories have been defined in
> enriched contexts?
> 
> I have an idea of how they can be defined in some particular
> contexts, such as Cat-categories or Simplicial-categories, but I
> don't know if there is a general definition or even if a
> definition in the above two contexts can be found in the
> literature.
> 
> Thanks
> 
> Manuel Bullejos
> 
> 
> 

Re: Comma categories

[Ross Street]

>Does any body know if comma categories have been defined in
>enriched contexts?

Lawvere's La Jolla paper, where general comma categories were introduced,
showed how to construct them from pullbacks and a "cylinder" (or "arrow
object") construction.  John Gray (SLNM p. 254) showed that cylinder is a
universal notion which a 2-category may or may not have. I pointed out
[Fibrations and Yoneda's lemma in a 2-category, Lecture Notes in Math. 420
(1974) 104-133;     MR53#585] that finite completeness for a 2-category
should mean that it have pullbacks, a terminal object, and cylinders (a
similar idea was in my PhD thesis for differential graded categories which
are finitely complete when they admit pullbacks, a zero object and
"suspension"). Finite completeness for 2-categories is further analysed in
[Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra 8
(1976) 149-181; MR53#5695].

More generally, finite completeness for a V-category  A (= a category with
homs enriched in  V) means that its underlying category has finite ordinary
limits, which are preserved by representables  A(a,-)  into  V,  and that
it admits cotensoring by the "finite" objects of  V.  There is some choice
about what you mean by "finite" object in  V  however "finitely
presentable" is often the right thing.  Sometimes, as in the case of  V =
Cat, the finite objects are generated by a few finite objects - that is why
"cylinder" plays the important role in 2-categories (it is the finite
generating object, cotensor with which is cylinder).

So why am I going on about finite limits in 2-categories?  Well, Lawvere's
construction shows that comma objects exist in any finitely complete
2-category.  Comma objects are particular finite limits just like
pullbacks.

In particular, there is a 2-category  V-Cat  of V-categories, V-functors
and V-natural transformations.  It is certainly complete (as a 2-category)
for any decent  V.   So, indeed, it is well known that comma objects (or
comma V-categories) exist.  They have their uses but NOT for the wonderful
use that Lawvere put them to:  Lawvere provided a formula for left (right)
Kan extensions of ordinary functors which involves taking a colimit (limit)
over a comma category. [Indeed, more is true; see my definition of
"pointwise Kan extension" in "Fibrations and Yoneda's lemma in a
2-category".]  However, this formula does not work even for additive
categories (= categories enriched in the monoidal category of abelian
groups).

Regards,
Ross
http://www.mpce.mq.edu.au/~street/

Re: Comma categories

[Vaughan Pratt]

>Does any body know if comma categories have been defined in
>enriched contexts?

I'm not sure if it's what you have in mind, but combining comma categories
and enrichment is the theme of

Casley, R.T., Crew, R.F., Meseguer, J., and Pratt, V.R., ``Temporal
Structures'', Mathematical Structures in Computer Science, Volume 1:2,
179-213, July 1991.

The abstract is at my web page as

	http://boole.stanford.edu/chuguide.html#P2

Vaughan Pratt

Comma categories

[Manuel Bullejos]

Does any body know if comma categories have been defined in
enriched contexts?

I have an idea of how they can be defined in some particular
contexts, such as Cat-categories or Simplicial-categories, but I
don't know if there is a general definition or even if a
definition in the above two contexts can be found in the
literature.

Thanks

Manuel Bullejos