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Suppose that $mathcalC$ is pointed…$mathcalC$ is equivalent to its category of pointed objects $mathcalC^ast/$ ?
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Definition 1. A category is called pointed if it has a zero object, i.e. if it has an initial object and a terminal object and they are isomorphic.
Definition 2. For $mathcalC$ a category with terminal object $ast$, the coslice category $mathcalC^ast/$ is the corresponding category of pointed objects: its
objects... are morphisms in $mathcalC$ of the form $ast oversetxto X$ (hence an object $X$ equipped with a choice of point; i.e. pointed object).
morphisms... in $mathcalC$ which preserve the chosen points.
Suppose that $mathcalC$ is pointed.
$mathcalC$ is equivalent to its category of pointed objects $mathcalC^ast/$ ?
What is the relationship between these two definitions?
category-theory
asked yesterday
Andres Felipe RamÃrez
466
 |Â
show 3 more comments
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0
down vote
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Definition 1. A category is called pointed if it has a zero object, i.e. if it has an initial object and a terminal object and they are isomorphic.
Definition 2. For $mathcalC$ a category with terminal object $ast$, the coslice category $mathcalC^ast/$ is the corresponding category of pointed objects: its
objects... are morphisms in $mathcalC$ of the form $ast oversetxto X$ (hence an object $X$ equipped with a choice of point; i.e. pointed object).
morphisms... in $mathcalC$ which preserve the chosen points.
Suppose that $mathcalC$ is pointed.
$mathcalC$ is equivalent to its category of pointed objects $mathcalC^ast/$ ?
What is the relationship between these two definitions?
category-theory
asked yesterday
Andres Felipe RamÃrez
466
1. Yes, this follows immediately from the definition, as $bullet$ is initial. 2. Every coslice category is pointed, with $bullet$ as the zero object.
– Qiaochu Yuan
yesterday
Is it to say that in any pointy category, does the definition of smash product (tensorail product) make sense?...This is wonderful!
– Andres Felipe RamÃrez
yesterday
I don't understand the connection between that question and anything that has been previously said, but see ncatlab.org/nlab/show/smash+product#ForGeneralPointedObjects.
– Qiaochu Yuan
yesterday
@QiaochuYuan... thanks!.... Excuse me ... I'm "thinking out loud"... I barely realize that in any category with zero object you can define a tensor product ...
– Andres Felipe RamÃrez
yesterday
I don't know what you mean by that. The construction I linked to takes as input an existing monoidal structure.
– Qiaochu Yuan
yesterday
 |Â
show 3 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Definition 1. A category is called pointed if it has a zero object, i.e. if it has an initial object and a terminal object and they are isomorphic.
Definition 2. For $mathcalC$ a category with terminal object $ast$, the coslice category $mathcalC^ast/$ is the corresponding category of pointed objects: its
objects... are morphisms in $mathcalC$ of the form $ast oversetxto X$ (hence an object $X$ equipped with a choice of point; i.e. pointed object).
morphisms... in $mathcalC$ which preserve the chosen points.
Suppose that $mathcalC$ is pointed.
$mathcalC$ is equivalent to its category of pointed objects $mathcalC^ast/$ ?
What is the relationship between these two definitions?
category-theory
asked yesterday
Andres Felipe RamÃrez
466
Definition 1. A category is called pointed if it has a zero object, i.e. if it has an initial object and a terminal object and they are isomorphic.
Definition 2. For $mathcalC$ a category with terminal object $ast$, the coslice category $mathcalC^ast/$ is the corresponding category of pointed objects: its
objects... are morphisms in $mathcalC$ of the form $ast oversetxto X$ (hence an object $X$ equipped with a choice of point; i.e. pointed object).
morphisms... in $mathcalC$ which preserve the chosen points.
Suppose that $mathcalC$ is pointed.
$mathcalC$ is equivalent to its category of pointed objects $mathcalC^ast/$ ?
What is the relationship between these two definitions?
category-theory
asked yesterday
Andres Felipe RamÃrez
466
asked yesterday
Andres Felipe RamÃrez
466
asked yesterday
Andres Felipe RamÃrez
466
asked yesterday
Andres Felipe RamÃrez
466
466
1. Yes, this follows immediately from the definition, as $bullet$ is initial. 2. Every coslice category is pointed, with $bullet$ as the zero object.
– Qiaochu Yuan
yesterday
Is it to say that in any pointy category, does the definition of smash product (tensorail product) make sense?...This is wonderful!
– Andres Felipe RamÃrez
yesterday
I don't understand the connection between that question and anything that has been previously said, but see ncatlab.org/nlab/show/smash+product#ForGeneralPointedObjects.
– Qiaochu Yuan
yesterday
@QiaochuYuan... thanks!.... Excuse me ... I'm "thinking out loud"... I barely realize that in any category with zero object you can define a tensor product ...
– Andres Felipe RamÃrez
yesterday
I don't know what you mean by that. The construction I linked to takes as input an existing monoidal structure.
– Qiaochu Yuan
yesterday
 |Â
show 3 more comments
1. Yes, this follows immediately from the definition, as $bullet$ is initial. 2. Every coslice category is pointed, with $bullet$ as the zero object.
– Qiaochu Yuan
yesterday
Is it to say that in any pointy category, does the definition of smash product (tensorail product) make sense?...This is wonderful!
– Andres Felipe RamÃrez
yesterday
I don't understand the connection between that question and anything that has been previously said, but see ncatlab.org/nlab/show/smash+product#ForGeneralPointedObjects.
– Qiaochu Yuan
yesterday
@QiaochuYuan... thanks!.... Excuse me ... I'm "thinking out loud"... I barely realize that in any category with zero object you can define a tensor product ...
– Andres Felipe RamÃrez
yesterday
I don't know what you mean by that. The construction I linked to takes as input an existing monoidal structure.
– Qiaochu Yuan
yesterday
1. Yes, this follows immediately from the definition, as $bullet$ is initial. 2. Every coslice category is pointed, with $bullet$ as the zero object.
– Qiaochu Yuan
yesterday
1. Yes, this follows immediately from the definition, as $bullet$ is initial. 2. Every coslice category is pointed, with $bullet$ as the zero object.
– Qiaochu Yuan
yesterday
Is it to say that in any pointy category, does the definition of smash product (tensorail product) make sense?...This is wonderful!
– Andres Felipe RamÃrez
yesterday
Is it to say that in any pointy category, does the definition of smash product (tensorail product) make sense?...This is wonderful!
– Andres Felipe RamÃrez
yesterday
I don't understand the connection between that question and anything that has been previously said, but see ncatlab.org/nlab/show/smash+product#ForGeneralPointedObjects.
– Qiaochu Yuan
yesterday
I don't understand the connection between that question and anything that has been previously said, but see ncatlab.org/nlab/show/smash+product#ForGeneralPointedObjects.
– Qiaochu Yuan
yesterday
@QiaochuYuan... thanks!.... Excuse me ... I'm "thinking out loud"... I barely realize that in any category with zero object you can define a tensor product ...
– Andres Felipe RamÃrez
yesterday
@QiaochuYuan... thanks!.... Excuse me ... I'm "thinking out loud"... I barely realize that in any category with zero object you can define a tensor product ...
– Andres Felipe RamÃrez
yesterday
I don't know what you mean by that. The construction I linked to takes as input an existing monoidal structure.
– Qiaochu Yuan
yesterday
I don't know what you mean by that. The construction I linked to takes as input an existing monoidal structure.
– Qiaochu Yuan
yesterday
 |Â
show 3 more comments
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1. Yes, this follows immediately from the definition, as $bullet$ is initial. 2. Every coslice category is pointed, with $bullet$ as the zero object.
– Qiaochu Yuan
yesterday
Is it to say that in any pointy category, does the definition of smash product (tensorail product) make sense?...This is wonderful!
– Andres Felipe RamÃrez
yesterday
I don't understand the connection between that question and anything that has been previously said, but see ncatlab.org/nlab/show/smash+product#ForGeneralPointedObjects.
– Qiaochu Yuan
yesterday
@QiaochuYuan... thanks!.... Excuse me ... I'm "thinking out loud"... I barely realize that in any category with zero object you can define a tensor product ...
– Andres Felipe RamÃrez
yesterday
I don't know what you mean by that. The construction I linked to takes as input an existing monoidal structure.
– Qiaochu Yuan
yesterday