A category-like structure without composition?Is there a free digraph associated to a graph?Can the inner structure of an object be systematically deduced from its position in the category?Do non-associative objects have a natural notion of representation?Why does Hom need an identity in the definition of the category?Sets = structured sets without structureWhat is the composition in SesquiAlg?What kind of category is a cyclically ordered set?random category theoryA model category of abelian categories?Cartesian liftings in double categories
A category-like structure without composition?
Is there a free digraph associated to a graph?Can the inner structure of an object be systematically deduced from its position in the category?Do non-associative objects have a natural notion of representation?Why does Hom need an identity in the definition of the category?Sets = structured sets without structureWhat is the composition in SesquiAlg?What kind of category is a cyclically ordered set?random category theoryA model category of abelian categories?Cartesian liftings in double categories
$begingroup$
Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!
reference-request ct.category-theory
asked Apr 3 at 22:28
APRAPR
995
$endgroup$
add a comment |
$begingroup$
Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!
reference-request ct.category-theory
asked Apr 3 at 22:28
APRAPR
995
$endgroup$
9
$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
Apr 3 at 23:10
3
$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
Apr 4 at 0:05
1
$begingroup$
Er -- I should clarify that 2-Segal spaces are a bit more general -- in addition to allowing composition to be undefined, they also allow composition to be multiply-defined. So they're like a category enriched in spans rather than pointed sets.
$endgroup$
– Tim Campion
Apr 4 at 18:52
add a comment |
$begingroup$
Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!
reference-request ct.category-theory
asked Apr 3 at 22:28
APRAPR
995
$endgroup$
Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!
reference-request ct.category-theory
reference-request ct.category-theory
asked Apr 3 at 22:28
APRAPR
995
asked Apr 3 at 22:28
APRAPR
995
edited Apr 3 at 23:36
APR
asked Apr 3 at 22:28
APRAPR
995
asked Apr 3 at 22:28
APRAPR
995
asked Apr 3 at 22:28
APRAPR
995
995
9
$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
Apr 3 at 23:10
3
$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
Apr 4 at 0:05
1
$begingroup$
Er -- I should clarify that 2-Segal spaces are a bit more general -- in addition to allowing composition to be undefined, they also allow composition to be multiply-defined. So they're like a category enriched in spans rather than pointed sets.
$endgroup$
– Tim Campion
Apr 4 at 18:52
add a comment |
9
$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
Apr 3 at 23:10
3
$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
Apr 4 at 0:05
1
$begingroup$
Er -- I should clarify that 2-Segal spaces are a bit more general -- in addition to allowing composition to be undefined, they also allow composition to be multiply-defined. So they're like a category enriched in spans rather than pointed sets.
$endgroup$
– Tim Campion
Apr 4 at 18:52
9
9
$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
Apr 3 at 23:10
$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
Apr 3 at 23:10
3
3
$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
Apr 4 at 0:05
$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
Apr 4 at 0:05
1
1
$begingroup$
Er -- I should clarify that 2-Segal spaces are a bit more general -- in addition to allowing composition to be undefined, they also allow composition to be multiply-defined. So they're like a category enriched in spans rather than pointed sets.
$endgroup$
– Tim Campion
Apr 4 at 18:52
$begingroup$
Er -- I should clarify that 2-Segal spaces are a bit more general -- in addition to allowing composition to be undefined, they also allow composition to be multiply-defined. So they're like a category enriched in spans rather than pointed sets.
$endgroup$
– Tim Campion
Apr 4 at 18:52
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.
A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.
answered Apr 3 at 23:45
Mike ShulmanMike Shulman
37.9k487236
$endgroup$
add a comment |
$begingroup$
Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.
edited Apr 4 at 0:22
Dan Petersen
26.1k277142
answered Apr 3 at 23:46
Theo Johnson-FreydTheo Johnson-Freyd
29.8k882253
$endgroup$
3
$begingroup$
I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
$endgroup$
– Kevin Walker
Apr 4 at 12:46
1
$begingroup$
I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
$endgroup$
– Kevin Walker
Apr 4 at 12:48
$begingroup$
@Dan Petersen Thanks for the correction.
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:38
$begingroup$
@KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:42
$begingroup$
@TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
$endgroup$
– Hurkyl
Apr 5 at 8:44
|
show 2 more comments
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.
A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.
answered Apr 3 at 23:45
Mike ShulmanMike Shulman
37.9k487236
$endgroup$
add a comment |
$begingroup$
As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.
A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.
answered Apr 3 at 23:45
Mike ShulmanMike Shulman
37.9k487236
$endgroup$
add a comment |
$begingroup$
As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.
A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.
answered Apr 3 at 23:45
Mike ShulmanMike Shulman
37.9k487236
$endgroup$
As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.
A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.
answered Apr 3 at 23:45
Mike ShulmanMike Shulman
37.9k487236
answered Apr 3 at 23:45
Mike ShulmanMike Shulman
37.9k487236
answered Apr 3 at 23:45
Mike ShulmanMike Shulman
37.9k487236
answered Apr 3 at 23:45
Mike ShulmanMike Shulman
37.9k487236
37.9k487236
add a comment |
add a comment |
$begingroup$
Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.
edited Apr 4 at 0:22
Dan Petersen
26.1k277142
answered Apr 3 at 23:46
Theo Johnson-FreydTheo Johnson-Freyd
29.8k882253
$endgroup$
3
$begingroup$
I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
$endgroup$
– Kevin Walker
Apr 4 at 12:46
1
$begingroup$
I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
$endgroup$
– Kevin Walker
Apr 4 at 12:48
$begingroup$
@Dan Petersen Thanks for the correction.
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:38
$begingroup$
@KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:42
$begingroup$
@TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
$endgroup$
– Hurkyl
Apr 5 at 8:44
|
show 2 more comments
$begingroup$
Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.
edited Apr 4 at 0:22
Dan Petersen
26.1k277142
answered Apr 3 at 23:46
Theo Johnson-FreydTheo Johnson-Freyd
29.8k882253
$endgroup$
3
$begingroup$
I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
$endgroup$
– Kevin Walker
Apr 4 at 12:46
1
$begingroup$
I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
$endgroup$
– Kevin Walker
Apr 4 at 12:48
$begingroup$
@Dan Petersen Thanks for the correction.
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:38
$begingroup$
@KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:42
$begingroup$
@TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
$endgroup$
– Hurkyl
Apr 5 at 8:44
|
show 2 more comments
$begingroup$
Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.
edited Apr 4 at 0:22
Dan Petersen
26.1k277142
answered Apr 3 at 23:46
Theo Johnson-FreydTheo Johnson-Freyd
29.8k882253
$endgroup$
Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.
edited Apr 4 at 0:22
Dan Petersen
26.1k277142
answered Apr 3 at 23:46
Theo Johnson-FreydTheo Johnson-Freyd
29.8k882253
edited Apr 4 at 0:22
Dan Petersen
26.1k277142
edited Apr 4 at 0:22
Dan Petersen
26.1k277142
edited Apr 4 at 0:22
Dan Petersen
26.1k277142
26.1k277142
answered Apr 3 at 23:46
Theo Johnson-FreydTheo Johnson-Freyd
29.8k882253
answered Apr 3 at 23:46
Theo Johnson-FreydTheo Johnson-Freyd
29.8k882253
answered Apr 3 at 23:46
Theo Johnson-FreydTheo Johnson-Freyd
29.8k882253
29.8k882253
3
$begingroup$
I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
$endgroup$
– Kevin Walker
Apr 4 at 12:46
1
$begingroup$
I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
$endgroup$
– Kevin Walker
Apr 4 at 12:48
$begingroup$
@Dan Petersen Thanks for the correction.
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:38
$begingroup$
@KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:42
$begingroup$
@TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
$endgroup$
– Hurkyl
Apr 5 at 8:44
|
show 2 more comments
3
$begingroup$
I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
$endgroup$
– Kevin Walker
Apr 4 at 12:46
1
$begingroup$
I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
$endgroup$
– Kevin Walker
Apr 4 at 12:48
$begingroup$
@Dan Petersen Thanks for the correction.
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:38
$begingroup$
@KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:42
$begingroup$
@TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
$endgroup$
– Hurkyl
Apr 5 at 8:44
3
3
$begingroup$
I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
$endgroup$
– Kevin Walker
Apr 4 at 12:46
$begingroup$
I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
$endgroup$
– Kevin Walker
Apr 4 at 12:46
1
1
$begingroup$
I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
$endgroup$
– Kevin Walker
Apr 4 at 12:48
$begingroup$
I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
$endgroup$
– Kevin Walker
Apr 4 at 12:48
$begingroup$
@Dan Petersen Thanks for the correction.
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:38
$begingroup$
@Dan Petersen Thanks for the correction.
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:38
$begingroup$
@KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:42
$begingroup$
@KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
$endgroup$
– Theo Johnson-Freyd
Apr 5 at 2:42
$begingroup$
@TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
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– Hurkyl
Apr 5 at 8:44
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@TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
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– Hurkyl
Apr 5 at 8:44
|
show 2 more comments
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9
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Shouldn't this be equivalent to a category enriched over pointed sets?
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– Qiaochu Yuan
Apr 3 at 23:10
3
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There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
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– Tim Campion
Apr 4 at 0:05
1
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Er -- I should clarify that 2-Segal spaces are a bit more general -- in addition to allowing composition to be undefined, they also allow composition to be multiply-defined. So they're like a category enriched in spans rather than pointed sets.
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– Tim Campion
Apr 4 at 18:52