term for concrete category with initial morphism for each element












1














I need a term for a concrete category over Set with a distinguished object $i,$ such that the morphisms from $i$ correspond to elements of the other objects. More formally, for every object $o$ and every element $x in o,$ there is exactly one morphism $m_{o,x}: i to O.$ Is this an established concept, and, if so, what is the nomenclature.



Note that the terms initial and pointed already have other meanings in category theory.



The question arose while revising https://arxiv.org/abs/1801.05775






category-theory







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  • Hi, did you write that paper?
    – magma
    Dec 28 '18 at 14:34










  • Do you mean something like the one-point set in category Set ?
    – magma
    Dec 28 '18 at 14:35
















1














I need a term for a concrete category over Set with a distinguished object $i,$ such that the morphisms from $i$ correspond to elements of the other objects. More formally, for every object $o$ and every element $x in o,$ there is exactly one morphism $m_{o,x}: i to O.$ Is this an established concept, and, if so, what is the nomenclature.



Note that the terms initial and pointed already have other meanings in category theory.



The question arose while revising https://arxiv.org/abs/1801.05775






category-theory







share|cite|improve this question









New contributor




shmuel is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • Hi, did you write that paper?
    – magma
    Dec 28 '18 at 14:34










  • Do you mean something like the one-point set in category Set ?
    – magma
    Dec 28 '18 at 14:35














1












1








1


1





I need a term for a concrete category over Set with a distinguished object $i,$ such that the morphisms from $i$ correspond to elements of the other objects. More formally, for every object $o$ and every element $x in o,$ there is exactly one morphism $m_{o,x}: i to O.$ Is this an established concept, and, if so, what is the nomenclature.



Note that the terms initial and pointed already have other meanings in category theory.



The question arose while revising https://arxiv.org/abs/1801.05775






category-theory







share|cite|improve this question









New contributor




shmuel is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











I need a term for a concrete category over Set with a distinguished object $i,$ such that the morphisms from $i$ correspond to elements of the other objects. More formally, for every object $o$ and every element $x in o,$ there is exactly one morphism $m_{o,x}: i to O.$ Is this an established concept, and, if so, what is the nomenclature.



Note that the terms initial and pointed already have other meanings in category theory.



The question arose while revising https://arxiv.org/abs/1801.05775






category-theory




category-theory






share|cite|improve this question









New contributor




shmuel is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









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shmuel is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




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edited Dec 26 '18 at 19:54









user376343

2,8382822




2,8382822






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asked Dec 26 '18 at 19:47









shmuel

1062




1062




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shmuel is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • Hi, did you write that paper?
    – magma
    Dec 28 '18 at 14:34










  • Do you mean something like the one-point set in category Set ?
    – magma
    Dec 28 '18 at 14:35


















  • Hi, did you write that paper?
    – magma
    Dec 28 '18 at 14:34










  • Do you mean something like the one-point set in category Set ?
    – magma
    Dec 28 '18 at 14:35
















Hi, did you write that paper?
– magma
Dec 28 '18 at 14:34




Hi, did you write that paper?
– magma
Dec 28 '18 at 14:34












Do you mean something like the one-point set in category Set ?
– magma
Dec 28 '18 at 14:35




Do you mean something like the one-point set in category Set ?
– magma
Dec 28 '18 at 14:35










1 Answer
1






active

oldest

votes


















6














If what you're after is a concrete category where $U$ is representable, then such categories are called representably concrete.






share|cite|improve this answer





















  • What I was looking for is probably the term generator (or separator.), but the definitions in ncatlab.org/nlab/show/separator and en.wikipedia.org/wiki/Generator_(category_theory) don't match. I need the term as defined in Wiki. Also, is there a term for a category with a generator?
    – shmuel
    Dec 26 '18 at 21:23










  • @shmuel - Isn't the definition in one of those just the contrapositive of the other?
    – Malice Vidrine
    Dec 26 '18 at 22:28










  • No; the nlab definition doesn't require that there be arrows from the generator to every other object.
    – shmuel
    Dec 27 '18 at 20:19











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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









6














If what you're after is a concrete category where $U$ is representable, then such categories are called representably concrete.






share|cite|improve this answer





















  • What I was looking for is probably the term generator (or separator.), but the definitions in ncatlab.org/nlab/show/separator and en.wikipedia.org/wiki/Generator_(category_theory) don't match. I need the term as defined in Wiki. Also, is there a term for a category with a generator?
    – shmuel
    Dec 26 '18 at 21:23










  • @shmuel - Isn't the definition in one of those just the contrapositive of the other?
    – Malice Vidrine
    Dec 26 '18 at 22:28










  • No; the nlab definition doesn't require that there be arrows from the generator to every other object.
    – shmuel
    Dec 27 '18 at 20:19
















6














If what you're after is a concrete category where $U$ is representable, then such categories are called representably concrete.






share|cite|improve this answer





















  • What I was looking for is probably the term generator (or separator.), but the definitions in ncatlab.org/nlab/show/separator and en.wikipedia.org/wiki/Generator_(category_theory) don't match. I need the term as defined in Wiki. Also, is there a term for a category with a generator?
    – shmuel
    Dec 26 '18 at 21:23










  • @shmuel - Isn't the definition in one of those just the contrapositive of the other?
    – Malice Vidrine
    Dec 26 '18 at 22:28










  • No; the nlab definition doesn't require that there be arrows from the generator to every other object.
    – shmuel
    Dec 27 '18 at 20:19














6












6








6






If what you're after is a concrete category where $U$ is representable, then such categories are called representably concrete.






share|cite|improve this answer












If what you're after is a concrete category where $U$ is representable, then such categories are called representably concrete.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 26 '18 at 19:55









Ittay Weiss

63.6k6101183




63.6k6101183












  • What I was looking for is probably the term generator (or separator.), but the definitions in ncatlab.org/nlab/show/separator and en.wikipedia.org/wiki/Generator_(category_theory) don't match. I need the term as defined in Wiki. Also, is there a term for a category with a generator?
    – shmuel
    Dec 26 '18 at 21:23










  • @shmuel - Isn't the definition in one of those just the contrapositive of the other?
    – Malice Vidrine
    Dec 26 '18 at 22:28










  • No; the nlab definition doesn't require that there be arrows from the generator to every other object.
    – shmuel
    Dec 27 '18 at 20:19


















  • What I was looking for is probably the term generator (or separator.), but the definitions in ncatlab.org/nlab/show/separator and en.wikipedia.org/wiki/Generator_(category_theory) don't match. I need the term as defined in Wiki. Also, is there a term for a category with a generator?
    – shmuel
    Dec 26 '18 at 21:23










  • @shmuel - Isn't the definition in one of those just the contrapositive of the other?
    – Malice Vidrine
    Dec 26 '18 at 22:28










  • No; the nlab definition doesn't require that there be arrows from the generator to every other object.
    – shmuel
    Dec 27 '18 at 20:19
















What I was looking for is probably the term generator (or separator.), but the definitions in ncatlab.org/nlab/show/separator and en.wikipedia.org/wiki/Generator_(category_theory) don't match. I need the term as defined in Wiki. Also, is there a term for a category with a generator?
– shmuel
Dec 26 '18 at 21:23




What I was looking for is probably the term generator (or separator.), but the definitions in ncatlab.org/nlab/show/separator and en.wikipedia.org/wiki/Generator_(category_theory) don't match. I need the term as defined in Wiki. Also, is there a term for a category with a generator?
– shmuel
Dec 26 '18 at 21:23












@shmuel - Isn't the definition in one of those just the contrapositive of the other?
– Malice Vidrine
Dec 26 '18 at 22:28




@shmuel - Isn't the definition in one of those just the contrapositive of the other?
– Malice Vidrine
Dec 26 '18 at 22:28












No; the nlab definition doesn't require that there be arrows from the generator to every other object.
– shmuel
Dec 27 '18 at 20:19




No; the nlab definition doesn't require that there be arrows from the generator to every other object.
– shmuel
Dec 27 '18 at 20:19










shmuel is a new contributor. Be nice, and check out our Code of Conduct.










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