Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: biproduct

Bottom of Page

1 to 24 of 24

- CommentRowNumber1.
- CommentAuthorColin Tan
- CommentTimeJan 15th 2019
- PermaLink
Author: Colin Tan
Format: MarkdownItexAdded the contents of the canonical isomorphism induced by some non-canonical isomorphism as coming from Lack's proof. <a href="https://ncatlab.org/nlab/revision/diff/biproduct/32">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/32">v32</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

Added the contents of the canonical isomorphism induced by some non-canonical isomorphism as coming from Lack’s proof.

diff, v32, current
- CommentRowNumber2.
- CommentAuthorColin Tan
- CommentTimeApr 1st 2019
- PermaLink
Author: Colin Tan
Format: MarkdownItexAdded an explicit formula for the natural map from coproducts to products. <a href="https://ncatlab.org/nlab/revision/diff/biproduct/33">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/33">v33</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

Added an explicit formula for the natural map from coproducts to products.

diff, v33, current
- CommentRowNumber3.
- CommentAuthorTodd_Trimble
- CommentTimeApr 6th 2019
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexI consider that formula, particularly the usage of $\sqcup$ (or rather that big chunky $\coprod$ I see), an abuse of notation. We have a coproduct functor $-\sqcup -$ and that takes a pair of morphisms $f: a \to b$ and $c \to d$ to $f \sqcup g: a \sqcup c \to b \sqcup d$. If instead you have a pair of maps $f: a \to c$ and $g: b \to c$ and you want a formula for the induced map $a \sqcup b \to c$, then the correct formula for it would be the composite $$a \sqcup b \stackrel{f \sqcup g}{\to} c \sqcup c \stackrel{\nabla_c}{\to} c$$ where $\nabla_c$ is the codiagonal map. Thus not to be notated as $f \sqcup g: a \sqcup b \to c$. Note that this construction is dual to the following construction which takes given maps $f: a \to b, g: a \to c$ to $$a \stackrel{\Delta_a}{\to} a \times a \stackrel{f \times g}{\to} b \times c$$ which you denote as $(f, g): a \to b \times c$. Morally, it seems "unfair" to favor the product with a nice snappy notation like $(f, g)$ (the notation you use) while not doing the same, or otherwise spelling out a cumbersome formula, for the dual construction on the coproduct side. My own habit has been to use something like $(f, g): a \sqcup b \to c$ as notation for the coproduct construction and $\langle f, g \rangle: a \to b \times c$ for the product construction and not play favorites. I'd be just as happy with $[f, g]: a \sqcup b \to c$ for the coproduct. Following those conventions, a formula for the map $c_1 \sqcup c_2 \to c_1 \times c_2$ might be $$[\langle Id_{c_1}, 0_{1, 2}\rangle, \langle 0_{2, 1}, Id_{c_2}\rangle]$$ or, dually, $$\langle [Id_{c_1}, 0_{2, 1}], [0_{1, 2}, Id_{c_2}] \rangle$$ (or some such).

I consider that formula, particularly the usage of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⊔</mo></mrow><annotation encoding="application/x-tex">\sqcup</annotation></semantics></math>$ (or rather that big chunky $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.16667em" rspace="0.16667em">∐</mo></mrow><annotation encoding="application/x-tex">\coprod</annotation></semantics></math>$ I see), an abuse of notation. We have a coproduct functor $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.11111em" rspace="0em">−</mo><mo>⊔</mo><mo lspace="0.11111em" rspace="0em">−</mo></mrow><annotation encoding="application/x-tex">-\sqcup -</annotation></semantics></math>$ and that takes a pair of morphisms $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">f: a \to b</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>→</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">c \to d</annotation></semantics></math>$ to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>⊔</mo><mi>g</mi><mo>:</mo><mi>a</mi><mo>⊔</mo><mi>c</mi><mo>→</mo><mi>b</mi><mo>⊔</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">f \sqcup g: a \sqcup c \to b \sqcup d</annotation></semantics></math>$ . If instead you have a pair of maps $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f: a \to c</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">g: b \to c</annotation></semantics></math>$ and you want a formula for the induced map $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>⊔</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a \sqcup b \to c</annotation></semantics></math>$ , then the correct formula for it would be the composite
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>a</mi><mo>⊔</mo><mi>b</mi><mover><mo>→</mo><mrow><mi>f</mi><mo>⊔</mo><mi>g</mi></mrow></mover><mi>c</mi><mo>⊔</mo><mi>c</mi><mover><mo>→</mo><mrow><msub><mo>∇</mo> <mi>c</mi></msub></mrow></mover><mi>c</mi></mrow><annotation encoding="application/x-tex">a \sqcup b \stackrel{f \sqcup g}{\to} c \sqcup c \stackrel{\nabla_c}{\to} c</annotation></semantics></math>$
where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∇</mo> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\nabla_c</annotation></semantics></math>$ is the codiagonal map. Thus not to be notated as $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>⊔</mo><mi>g</mi><mo>:</mo><mi>a</mi><mo>⊔</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f \sqcup g: a \sqcup b \to c</annotation></semantics></math>$ .

Note that this construction is dual to the following construction which takes given maps $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>b</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f: a \to b, g: a \to c</annotation></semantics></math>$ to
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>a</mi><mover><mo>→</mo><mrow><msub><mi>Δ</mi> <mi>a</mi></msub></mrow></mover><mi>a</mi><mo>×</mo><mi>a</mi><mover><mo>→</mo><mrow><mi>f</mi><mo>×</mo><mi>g</mi></mrow></mover><mi>b</mi><mo>×</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a \stackrel{\Delta_a}{\to} a \times a \stackrel{f \times g}{\to} b \times c</annotation></semantics></math>$
which you denote as $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><mi>a</mi><mo>→</mo><mi>b</mi><mo>×</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">(f, g): a \to b \times c</annotation></semantics></math>$ . Morally, it seems “unfair” to favor the product with a nice snappy notation like $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f, g)</annotation></semantics></math>$ (the notation you use) while not doing the same, or otherwise spelling out a cumbersome formula, for the dual construction on the coproduct side. My own habit has been to use something like $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><mi>a</mi><mo>⊔</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">(f, g): a \sqcup b \to c</annotation></semantics></math>$ as notation for the coproduct construction and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">⟨</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">⟩</mo><mo>:</mo><mi>a</mi><mo>→</mo><mi>b</mi><mo>×</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\langle f, g \rangle: a \to b \times c</annotation></semantics></math>$ for the product construction and not play favorites. I’d be just as happy with $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">]</mo><mo>:</mo><mi>a</mi><mo>⊔</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">[f, g]: a \sqcup b \to c</annotation></semantics></math>$ for the coproduct.

Following those conventions, a formula for the map $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>⊔</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">c_1 \sqcup c_2 \to c_1 \times c_2</annotation></semantics></math>$ might be
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">⟨</mo><msub><mi>Id</mi> <mrow><msub><mi>c</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mn>0</mn> <mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">⟩</mo><mo>,</mo><mo stretchy="false">⟨</mo><msub><mn>0</mn> <mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>Id</mi> <mrow><msub><mi>c</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">⟩</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\langle Id_{c_1}, 0_{1, 2}\rangle, \langle 0_{2, 1}, Id_{c_2}\rangle]</annotation></semantics></math>$
or, dually,
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">⟨</mo><mo stretchy="false">[</mo><msub><mi>Id</mi> <mrow><msub><mi>c</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mn>0</mn> <mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mn>0</mn> <mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>Id</mi> <mrow><msub><mi>c</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle [Id_{c_1}, 0_{2, 1}], [0_{1, 2}, Id_{c_2}] \rangle</annotation></semantics></math>$
(or some such).
- CommentRowNumber4.
- CommentAuthorDavidRoberts
- CommentTimeApr 7th 2019
- PermaLink
Author: DavidRoberts
Format: MarkdownItexPerhaps ironically, I would use parentheses for maps to products (like vector notation for functions with Euclidean codomain) and angle brackets for maps out of a coproduct (which I've seen used before, but I can't recall where).

Perhaps ironically, I would use parentheses for maps to products (like vector notation for functions with Euclidean codomain) and angle brackets for maps out of a coproduct (which I’ve seen used before, but I can’t recall where).
- CommentRowNumber5.
- CommentAuthorTodd_Trimble
- CommentTimeApr 7th 2019
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexDavid, that way would be fine with me too!

David, that way would be fine with me too!
- CommentRowNumber6.
- CommentAuthorDavidRoberts
- CommentTimeApr 7th 2019
- PermaLink
Author: DavidRoberts
Format: MarkdownItexAnd yes, like Todd, I strongly disagree with the notation $f \coprod g$ for maps out of a coproduct. Just like one shouldn't write $f\times g$ for a map to a product.

And yes, like Todd, I strongly disagree with the notation $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo lspace="0.16667em" rspace="0.16667em">∐</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f \coprod g</annotation></semantics></math>$ for maps out of a coproduct. Just like one shouldn’t write $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>×</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f\times g</annotation></semantics></math>$ for a map to a product.
- CommentRowNumber7.
- CommentAuthorMike Shulman
- CommentTimeApr 7th 2019
- PermaLink
Author: Mike Shulman
Format: MarkdownItexMe three. I use $(f,g)$ for map into a product (analogously to the standard notation for *elements* of a cartesian product set, which it generalizes if you identify elements with maps out of 1) and usually $[f,g]$ for maps out of a coproduct, calling them "pairing" and "copairing" respectively.

Me three. I use $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g)</annotation></semantics></math>$ for map into a product (analogously to the standard notation for elements of a cartesian product set, which it generalizes if you identify elements with maps out of 1) and usually $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[f,g]</annotation></semantics></math>$ for maps out of a coproduct, calling them “pairing” and “copairing” respectively.
- CommentRowNumber8.
- CommentAuthorDavidRoberts
- CommentTimeApr 7th 2019
- PermaLink
Author: DavidRoberts
Format: MarkdownItexI also like the version Mike has for copairing, FWIW.

I also like the version Mike has for copairing, FWIW.
- CommentRowNumber9.
- CommentAuthorDmitri Pavlov
- CommentTimeApr 8th 2019
- PermaLink
Author: Dmitri Pavlov
Format: MarkdownItexI also disagree with f∐g and I like [f,g]. Additionally, ∐ (\coprod) instead of ⊔ (\sqcup) is a wrong symbol to use for binary coproducts, just like ⨁ (\bigoplus) instead of ⊕ (\oplus) is a wrong symbol for binary direct sums.

I also disagree with f∐g and I like [f,g].

Additionally, ∐ (\coprod) instead of ⊔ (\sqcup) is a wrong symbol to use for binary coproducts, just like ⨁ (\bigoplus) instead of ⊕ (\oplus) is a wrong symbol for binary direct sums.
- CommentRowNumber10.
- CommentAuthorMike Shulman
- CommentTimeApr 8th 2019
- PermaLink
Author: Mike Shulman
Format: MarkdownItexNote that in addition to $\coprod$ (`\coprod`) and $\sqcup$ (`\sqcup`) there is also $\amalg$ (`\amalg`). I've never been clear on whether there is an intended semantic distinction between $\sqcup$ and $\amalg$, but both seem to be sized as binary operators.

Note that in addition to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.16667em" rspace="0.16667em">∐</mo></mrow><annotation encoding="application/x-tex">\coprod</annotation></semantics></math>$ (\coprod) and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⊔</mo></mrow><annotation encoding="application/x-tex">\sqcup</annotation></semantics></math>$ (\sqcup) there is also $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⨿</mo></mrow><annotation encoding="application/x-tex">\amalg</annotation></semantics></math>$ (\amalg). I’ve never been clear on whether there is an intended semantic distinction between $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⊔</mo></mrow><annotation encoding="application/x-tex">\sqcup</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⨿</mo></mrow><annotation encoding="application/x-tex">\amalg</annotation></semantics></math>$ , but both seem to be sized as binary operators.
- CommentRowNumber11.
- CommentAuthorTodd_Trimble
- CommentTimeApr 8th 2019
- (edited Apr 8th 2019)
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexFollowing this discussion, and unless there are serious objections, some time soon I am going to start making edits at a number of pages, getting around to [[biproduct]] after clearing away some underbrush elsewhere. I've made a start at [[product]], and following Dmitri's observation, I would like to edit away in [[coproduct]] the big chunky $\coprod$ as a binary operator, replacing it with $\sqcup$. (Personally, in my own work I prefer to use $\sum$ over $\coprod$, and $+$ over $\sqcup$, but it would be close to impossible to get universal agreement on such things. However, some mention of these alternatives should be made.) I would guess that people like '\amalg' for pushout notation like $Y \amalg_X Z$. Again, my habit is usually to write $Y +_X Z$.

Following this discussion, and unless there are serious objections, some time soon I am going to start making edits at a number of pages, getting around to biproduct after clearing away some underbrush elsewhere. I’ve made a start at product, and following Dmitri’s observation, I would like to edit away in coproduct the big chunky $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.16667em" rspace="0.16667em">∐</mo></mrow><annotation encoding="application/x-tex">\coprod</annotation></semantics></math>$ as a binary operator, replacing it with $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⊔</mo></mrow><annotation encoding="application/x-tex">\sqcup</annotation></semantics></math>$ .

(Personally, in my own work I prefer to use $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.16667em" rspace="0.16667em">∑</mo></mrow><annotation encoding="application/x-tex">\sum</annotation></semantics></math>$ over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.16667em" rspace="0.16667em">∐</mo></mrow><annotation encoding="application/x-tex">\coprod</annotation></semantics></math>$ , and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.11111em" rspace="0em">+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math>$ over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⊔</mo></mrow><annotation encoding="application/x-tex">\sqcup</annotation></semantics></math>$ , but it would be close to impossible to get universal agreement on such things. However, some mention of these alternatives should be made.)

I would guess that people like ’\amalg’ for pushout notation like $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi><msub><mo>⨿</mo> <mi>X</mi></msub><mi>Z</mi></mrow><annotation encoding="application/x-tex">Y \amalg_X Z</annotation></semantics></math>$ . Again, my habit is usually to write $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi><msub><mo>+</mo> <mi>X</mi></msub><mi>Z</mi></mrow><annotation encoding="application/x-tex">Y +_X Z</annotation></semantics></math>$ .
- CommentRowNumber12.
- CommentAuthorMike Shulman
- CommentTimeApr 8th 2019
- PermaLink
Author: Mike Shulman
Format: MarkdownItexThanks Todd! I use $+$ sometimes too, but particularly in extensive categories where the coproduct really is a "disjoint union" I often find $\sqcup$ or $\amalg$ easier for me personally to parse, and I also sometimes find that a $+$ isn't quite strong enough to comfortably support subscripts as used for pushouts (particularly when the subscript gets longer than a single letter). And when both a category and its internal type theory are in play, I sometimes find it's useful to distinguish between $\coprod$ for the external coproducts and $\sum$ for the internal $\Sigma$-types.

Thanks Todd!

I use $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.11111em" rspace="0em">+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math>$ sometimes too, but particularly in extensive categories where the coproduct really is a “disjoint union” I often find $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⊔</mo></mrow><annotation encoding="application/x-tex">\sqcup</annotation></semantics></math>$ or $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⨿</mo></mrow><annotation encoding="application/x-tex">\amalg</annotation></semantics></math>$ easier for me personally to parse, and I also sometimes find that a $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.11111em" rspace="0em">+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math>$ isn’t quite strong enough to comfortably support subscripts as used for pushouts (particularly when the subscript gets longer than a single letter). And when both a category and its internal type theory are in play, I sometimes find it’s useful to distinguish between $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.16667em" rspace="0.16667em">∐</mo></mrow><annotation encoding="application/x-tex">\coprod</annotation></semantics></math>$ for the external coproducts and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.16667em" rspace="0.16667em">∑</mo></mrow><annotation encoding="application/x-tex">\sum</annotation></semantics></math>$ for the internal $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>$ -types.
- CommentRowNumber13.
- CommentAuthorDmitri Pavlov
- CommentTimeMar 31st 2020
- PermaLink
Author: Dmitri Pavlov
Format: MarkdownItexAdded a pointless definition. <a href="https://ncatlab.org/nlab/revision/diff/biproduct/34">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/34">v34</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

Added a pointless definition.

diff, v34, current
- CommentRowNumber14.
- CommentAuthornLab edit announcer
- CommentTimeJun 29th 2021
- PermaLink
Author: nLab edit announcer
Format: MarkdownItexI have added several more elementary and perhaps more enlightening examples. I think the example that was there was meant as an entry for the more general context of "ambidexterity", and it somehow ended up being the only example for this very elementary concept. Anonymous <a href="https://ncatlab.org/nlab/revision/diff/biproduct/38">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/38">v38</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

I have added several more elementary and perhaps more enlightening examples. I think the example that was there was meant as an entry for the more general context of “ambidexterity”, and it somehow ended up being the only example for this very elementary concept.

Anonymous

diff, v38, current
- CommentRowNumber15.
- CommentAuthorUrs
- CommentTimeJun 29th 2021
- PermaLink
Author: Urs
Format: MarkdownItexadded more hyperlinks to keywords in these examples (*[[Ab]]*, *[[derived category]]*, *[[stable homotopy category]]*, *[[exact functor]]*) <a href="https://ncatlab.org/nlab/revision/diff/biproduct/39">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/39">v39</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

added more hyperlinks to keywords in these examples (Ab, derived category, stable homotopy category, exact functor)

diff, v39, current
- CommentRowNumber16.
- CommentAuthornLab edit announcer
- CommentTimeJun 18th 2022
- PermaLink
Author: nLab edit announcer
Format: MarkdownItexadd link to Karvonen's page Antonin Delpeuch <a href="https://ncatlab.org/nlab/revision/diff/biproduct/42">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/42">v42</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

add link to Karvonen’s page

Antonin Delpeuch

diff, v42, current
- CommentRowNumber17.
- CommentAuthorJ-B Vienney
- CommentTimeNov 24th 2022
- (edited Nov 24th 2022)
- PermaLink
Author: J-B Vienney
Format: MarkdownItexPut examples of categories with binary products and coproducts which are not equal and don't give biproducts. <a href="https://ncatlab.org/nlab/revision/diff/biproduct/45">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/45">v45</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

Put examples of categories with binary products and coproducts which are not equal and don’t give biproducts.

diff, v45, current
- CommentRowNumber18.
- CommentAuthorJ-B Vienney
- CommentTimeNov 24th 2022
- PermaLink
Author: J-B Vienney
Format: MarkdownItexAdded [[Rel]] as an example of category with biproducts. <a href="https://ncatlab.org/nlab/revision/diff/biproduct/45">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/45">v45</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

Added Rel as an example of category with biproducts.

diff, v45, current
- CommentRowNumber19.
- CommentAuthorJ-B Vienney
- CommentTimeNov 24th 2022
- PermaLink
Author: J-B Vienney
Format: MarkdownItexAdded the example of sup-lattices <a href="https://ncatlab.org/nlab/revision/diff/biproduct/46">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/46">v46</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

Added the example of sup-lattices

diff, v46, current
- CommentRowNumber20.
- CommentAuthorJ-B Vienney
- CommentTimeFeb 5th 2023
- PermaLink
Author: J-B Vienney
Format: MarkdownItexAdded the usual definition in an additive category <a href="https://ncatlab.org/nlab/revision/diff/biproduct/47">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/47">v47</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

Added the usual definition in an additive category

diff, v47, current
- CommentRowNumber21.
- CommentAuthorUrs
- CommentTimeFeb 5th 2023
- PermaLink
Author: Urs
Format: MarkdownItexturned the three Definition-sections into subsections of a single Definition-section <a href="https://ncatlab.org/nlab/revision/diff/biproduct/48">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/48">v48</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

turned the three Definition-sections into subsections of a single Definition-section

diff, v48, current
- CommentRowNumber22.
- CommentAuthormattecapu
- CommentTimeAug 25th 2023
- PermaLink
Author: mattecapu
Format: MarkdownItexAdded remark that semiadditive categories are (co)cartesian objects in the 2-category of (co)cartesian categories and (co)product-preserving functors. <a href="https://ncatlab.org/nlab/revision/diff/biproduct/49">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/49">v49</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

Added remark that semiadditive categories are (co)cartesian objects in the 2-category of (co)cartesian categories and (co)product-preserving functors.

diff, v49, current
- CommentRowNumber23.
- CommentAuthorMike Shulman
- CommentTimeFeb 23rd 2024
- PermaLink
Author: Mike Shulman
Format: MarkdownItexAdd citations to generalizations of Houston's result <a href="https://ncatlab.org/nlab/revision/diff/biproduct/52">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/52">v52</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

Add citations to generalizations of Houston’s result

diff, v52, current
- CommentRowNumber24.
- CommentAuthorJ-B Vienney
- CommentTimeMar 6th 2024
- PermaLink
Author: J-B Vienney
Format: MarkdownItexAdded the definition of a biproduct of $n \ge 3$ objects in an additive category which requires one more equation than in the case $n=2$ (namely $i_k;p_l=0$ if $k \neq l$, which is automatically verified for $n=2$). <a href="https://ncatlab.org/nlab/revision/diff/biproduct/53">diff</a>, <a href="https://ncatlab.org/nlab/revision/biproduct/53">v53</a>, <a href="https://ncatlab.org/nlab/show/biproduct">current</a>

Added the definition of a biproduct of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n \ge 3</annotation></semantics></math>$ objects in an additive category which requires one more equation than in the case $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math>$ (namely $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>i</mi> <mi>k</mi></msub><mo>;</mo><msub><mi>p</mi> <mi>l</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">i_k;p_l=0</annotation></semantics></math>$ if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≠</mo><mi>l</mi></mrow><annotation encoding="application/x-tex">k \neq l</annotation></semantics></math>$ , which is automatically verified for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math>$ ).

diff, v53, current

1 to 24 of 24

nForum

Discussion Feed

Not signed in

Site Tag Cloud

nLab > Latest Changes: biproduct