* Name for not-quite-additive categories @ 2012-01-06 19:04 Michael Barr 2012-01-07 2:02 ` George Janelidze ` (3 more replies) 0 siblings, 4 replies; 5+ messages in thread From: Michael Barr @ 2012-01-06 19:04 UTC (permalink / raw) To: Categories list Has anyone settled on a term to describe categories (such as commutative monoids) in which finite sums and products coincide but are not quite additive? I guess they are commutative monoid enriched. Michael -- Any society that would give up a little liberty to gain a little security will deserve neither and lose both. Benjamin Franklin [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: Name for not-quite-additive categories 2012-01-06 19:04 Name for not-quite-additive categories Michael Barr @ 2012-01-07 2:02 ` George Janelidze 2012-01-07 8:02 ` Robin Houston ` (2 subsequent siblings) 3 siblings, 0 replies; 5+ messages in thread From: George Janelidze @ 2012-01-07 2:02 UTC (permalink / raw) To: Michael Barr, Categories list Everything I say below is known, old, and contains no results that are mine. In fact it goes back to Mac Lane's "Duality for Groups", even though not everything is explicit there. So, I omitted proofs - but I shall gladly recall any of them if anyone is interested. 1. Let us call a category pointed if it is enriched in POINTED SETS. Such an enrichment is unique if it exists - even if the category does not have initial (=terminal) object. 2. When C is pointed, for every two objects X and Y in C, we can obviously define the canonical morphism I : X+Y --> XxY from the coproduct X+Y to the product XxY (assuming or not that + and x are "chosen"). When C has binary products and binary coproducts we say that they coincide if all such canonical morphisms are isomorphisms. 3. The following conditions on a pointed category C are equivalent: (a) C has binary products and binary coproducts (let us assume "not chosen"), and they coincide; (b) C has binary products and, for every product diagram X <--p-- Z --q--> Y, there exist morphisms i : X --> Z and j : Y --> Z forming a coproduct diagram and satisfying the equalities pi = 1, qj = 1, pj = 0, qi = 0. (c) C has binary products and admits an enrichment in COMMUTATIVE MONOIDS. (d) C has binary products and admits a unique enrichment in COMMUTATIVE MONOIDS. (e) C admits an enrichment in COMMUTATIVE MONOIDS and, for every two objects X and Y in C, there exist a diagram of the form X <--p-- Z --q--> Y --i--> <--j-- with pi = 1, qj = 1, pj = 0, qi = 0, and ip+jq = 1. (One can also add the conditions dual to (b)-(d), require uniqueness of the enrichment in (e), and deduce associativity and commutativity of the "hom-unitary-magmas" from other axioms of course). As I understand, the question is, how to call a pointed category with finite products satisfying the equivalent conditions (a)-(e) above. For instance Freyd and Scedrov call it "half-additive" in their book, and I don't know any better name. George -------------------------------------------------- From: "Michael Barr" <barr@math.mcgill.ca> Sent: Friday, January 06, 2012 9:04 PM To: "Categories list" <categories@mta.ca> Subject: categories: Name for not-quite-additive categories > Has anyone settled on a term to describe categories (such as commutative > monoids) in which finite sums and products coincide but are not quite > additive? I guess they are commutative monoid enriched. > > Michael > > -- > Any society that would give up a little liberty to gain a little > security will deserve neither and lose both. > > Benjamin Franklin [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: Name for not-quite-additive categories 2012-01-06 19:04 Name for not-quite-additive categories Michael Barr 2012-01-07 2:02 ` George Janelidze @ 2012-01-07 8:02 ` Robin Houston 2012-01-07 8:04 ` rlk 2012-01-07 11:29 ` Prof. Peter Johnstone 3 siblings, 0 replies; 5+ messages in thread From: Robin Houston @ 2012-01-07 8:02 UTC (permalink / raw) To: Michael Barr; +Cc: Categories list The term semiadditive has been used for at least 45 years[1], and is still in use today[2]. It's not used all that often, but it has a good pedigree and is unambiguous. Robin 1. Barry Mitchell, Theory of Categories, 1965, section 1.18 2. http://ncatlab.org/nlab/show/biproduct#semiadditive_categories_11 On 6 Jan 2012, at 19:04, Michael Barr <barr@math.mcgill.ca> wrote: > Has anyone settled on a term to describe categories (such as commutative > monoids) in which finite sums and products coincide but are not quite > additive? I guess they are commutative monoid enriched. > > Michael > > -- > Any society that would give up a little liberty to gain a little > security will deserve neither and lose both. > > Benjamin Franklin [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: Name for not-quite-additive categories 2012-01-06 19:04 Name for not-quite-additive categories Michael Barr 2012-01-07 2:02 ` George Janelidze 2012-01-07 8:02 ` Robin Houston @ 2012-01-07 8:04 ` rlk 2012-01-07 11:29 ` Prof. Peter Johnstone 3 siblings, 0 replies; 5+ messages in thread From: rlk @ 2012-01-07 8:04 UTC (permalink / raw) To: Michael Barr; +Cc: Categories list Michael Barr writes: > Has anyone settled on a term to describe categories (such as commutative > monoids) in which finite sums and products coincide but are not quite > additive? I guess they are commutative monoid enriched. > > Michael I've been writing of direct sums when sums and products coincide and then of categories with direct sums. -- Bob [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: Name for not-quite-additive categories 2012-01-06 19:04 Name for not-quite-additive categories Michael Barr ` (2 preceding siblings ...) 2012-01-07 8:04 ` rlk @ 2012-01-07 11:29 ` Prof. Peter Johnstone 3 siblings, 0 replies; 5+ messages in thread From: Prof. Peter Johnstone @ 2012-01-07 11:29 UTC (permalink / raw) To: Michael Barr; +Cc: Categories list "Semi-additive" seems a good enough name to me. Peter Johnstone On Fri, 6 Jan 2012, Michael Barr wrote: > Has anyone settled on a term to describe categories (such as commutative > monoids) in which finite sums and products coincide but are not quite > additive? I guess they are commutative monoid enriched. > > Michael > > -- > Any society that would give up a little liberty to gain a little > security will deserve neither and lose both. > > Benjamin Franklin [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
end of thread, other threads:[~2012-01-07 11:29 UTC | newest] Thread overview: 5+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2012-01-06 19:04 Name for not-quite-additive categories Michael Barr 2012-01-07 2:02 ` George Janelidze 2012-01-07 8:02 ` Robin Houston 2012-01-07 8:04 ` rlk 2012-01-07 11:29 ` Prof. Peter Johnstone
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