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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 13th 2011
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   Author: David_Corfield
   Format: MarkdownItexIs there a modern version of Mac Lane's bicategories, as define in his 1950 paper [Duality for Groups](http://www.ams.org/journals/bull/1950-56-06/S0002-9904-1950-09427-0/S0002-9904-1950-09427-0.pdf). >DEFINITION. A *bicategory* $\mathcal{C}$ is a category with two given subclasses of mappings, the classes of "injections" ($\kappa$) and "projections" ($\pi$) subject to the axioms BC-0 to BC-6 below. >BC-0. A mapping equal to an injection (projection) is itself an injection (projection). >BC-1. Every identity of $\mathcal{C}$ is both an injection and a projection. >BC-2. If the product of two injections (projections) is defined, it is an injection (projection). >BC-3. (Canonical decomposition). Every mapping $\alpha$ of the bicategory can be represented uniquely as a product $\alpha = \kappa \theta \pi$, in which $\kappa$ is an injection, $\theta$ an equivalence, and $\pi$ a projection. >Any mapping of the form $\lambda = \kappa \theta$ (that is, any mapping with $\pi$ equal to an identity in the canonical decomposition) is called a *submap*; any mapping of the form $\pho = \theta \pi$ is called a *supermap*. >BC-4. If the product of two submaps (supermaps) is defined, it is a submap (supermap). >Any product $\kappa_1 \pi_1...\kappa_n \pi_n$ of injections $\kappa_i$ and projections $\pi_i$ is called an idemmap. >BC-5. If two idemmaps have the same range and the same domain, they are equal. >BC-6. For each object $A$, the class of all injections with range $A$ is a set, and the class of all projections with domain $A$ is a set.

   Is there a modern version of Mac Lane’s bicategories, as define in his 1950 paper Duality for Groups.

   DEFINITION. A bicategory $\mathcal{C}$ is a category with two given subclasses of mappings, the classes of “injections” ($\kappa$) and “projections” ($\pi$) subject to the axioms BC-0 to BC-6 below.

   BC-0. A mapping equal to an injection (projection) is itself an injection (projection).

   BC-1. Every identity of $\mathcal{C}$ is both an injection and a projection.

   BC-2. If the product of two injections (projections) is defined, it is an injection (projection).

   BC-3. (Canonical decomposition). Every mapping $\alpha$ of the bicategory can be represented uniquely as a product $\alpha = \kappa \theta \pi$, in which $\kappa$ is an injection, $\theta$ an equivalence, and $\pi$ a projection.

   Any mapping of the form $\lambda = \kappa \theta$ (that is, any mapping with $\pi$ equal to an identity in the canonical decomposition) is called a submap; any mapping of the form $\pho = \theta \pi$ is called a supermap.

   BC-4. If the product of two submaps (supermaps) is defined, it is a submap (supermap).

   Any product $\kappa_1 \pi_1...\kappa_n \pi_n$ of injections $\kappa_i$ and projections $\pi_i$ is called an idemmap.

   BC-5. If two idemmaps have the same range and the same domain, they are equal.

   BC-6. For each object $A$, the class of all injections with range $A$ is a set, and the class of all projections with domain $A$ is a set.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 13th 2011
   • (edited Jan 13th 2011)
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   Author: David_Corfield
   Format: MarkdownItexAh ha! From [Homology and homotopy in semi-abelian categories](http://arxiv.org/abs/math/0607100) >For the results in this thesis, the ideal context is that of semi-abelian categories [78]: pointed, exact, protomodular with binary coproducts. Not only does it encompass all the mentioned categorical environments, so that all results in the foregoing sections are valid in a semi-abelian category; there is also a historical reason for the importance of this notion. Indeed, introducing semi-abelian categories, Janelidze, Marki and Tholen solved Mac Lane’s long standing problem [94] of finding a framework that reflects the categorical properties of non-abelian groups as nicely as abelian categories do for abelian groups. But over the years, many different people came up with partial solutions to this problem, proving theorems starting from various sets of axioms, which all require “good behaviour” of normal mono- and epimorphisms. In the paper [78], the relationship between these “oldstyle” axioms and the semi-abelian context is explained, and thus the old results are incorporated into the new theory. [94] is Duality for Groups. Finding the connections between "pointed, exact, protomodular with binary coproducts" and BC-0 to BC-6 above is not so obvious to me, however.

   Ah ha! From Homology and homotopy in semi-abelian categories

   For the results in this thesis, the ideal context is that of semi-abelian categories [78]: pointed, exact, protomodular with binary coproducts. Not only does it encompass all the mentioned categorical environments, so that all results in the foregoing sections are valid in a semi-abelian category; there is also a historical reason for the importance of this notion. Indeed, introducing semi-abelian categories, Janelidze, Marki and Tholen solved Mac Lane’s long standing problem [94] of finding a framework that reflects the categorical properties of non-abelian groups as nicely as abelian categories do for abelian groups. But over the years, many different people came up with partial solutions to this problem, proving theorems starting from various sets of axioms, which all require “good behaviour” of normal mono- and epimorphisms. In the paper [78], the relationship between these “oldstyle” axioms and the semi-abelian context is explained, and thus the old results are incorporated into the new theory.

   [94] is Duality for Groups.

   Finding the connections between “pointed, exact, protomodular with binary coproducts” and BC-0 to BC-6 above is not so obvious to me, however.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeJan 13th 2011
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   Author: zskoda
   Format: MarkdownItexI vaguely remember that the concept of bicategory from 1950 is an argument that MacLane had a concept of an [[abelian category]] well before Grothendieck's 1955 introduction in Kansas seminar (published later as [[Tohoku]]). I do not quite understand the argument.

   I vaguely remember that the concept of bicategory from 1950 is an argument that MacLane had a concept of an abelian category well before Grothendieck’s 1955 introduction in Kansas seminar (published later as Tohoku). I do not quite understand the argument.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJan 13th 2011
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   Author: Mike Shulman
   Format: MarkdownItexThe axioms given above just look to me like a [[unique factorization system]] plus some extra evil data.

   The axioms given above just look to me like a unique factorization system plus some extra evil data.

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 13th 2011
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   Author: DavidRoberts
   Format: MarkdownItexYeah, weird things like, if a map is equal to one that is in the class, then it is in the class. :S

   Yeah, weird things like, if a map is equal to one that is in the class, then it is in the class. :S