Mixing
A multicategory is a … with one object?
$begingroup$
We all know that
A monoidal category is a bicategory with one object.
How do we fill in the blank in the following sentence?
A multicategory is a ... with one object.
The answer is fairly clear: it'll be a bit like a bicategory, but instead of being able to compose $1$-cells straightforwardly, all we can do is specify, for any 'composable' sequence
$$
A_0 xrightarrow{f_1} cdots xrightarrow{f_n} A_n,,
$$
of $1$-cells and any $1$-cell $g colon A_0 to A_n$, the set of $2$-cells
$$
f_1,dots,f_n Rightarrow g,.
$$
What is the name of such a thing? A natural example, for a given SMCC $mathcal V$, has $mathcal V$-enriched categories as the objects, profunctors $mathcal C^{op}otimes mathcal Cto mathcal V$ as the $1$-cells and, for any collection
$$
mathcal C_0,cdots,mathcal C_n
$$
of categories and profunctors $F_icolon mathcal C_{i-1}^{op}otimes mathcal C_ito mathcal V$, $Gcolon mathcal C_0^{op}otimes mathcal C_n to mathcal V$, the set of extranatural transformations
$$
phi_{b_1,cdots,b_{n-1}} colon F_0(a,b_1) otimes cdots otimes F_n(b_{n-1},c)Rightarrow G(a,c)
$$
as the $2$-cells $F_1,cdots,F_n Rightarrow G$.
Of course, if $mathcal V$ is cocomplete, then this is equivalent to the usual bicategory of $mathcal V$-enriched profunctors.
reference-request ct.category-theory higher-category-theory enriched-category-theory profunctors
asked Jan 11 at 13:21
John GowersJohn Gowers
412213
$endgroup$
add a comment |
$begingroup$
We all know that
A monoidal category is a bicategory with one object.
How do we fill in the blank in the following sentence?
A multicategory is a ... with one object.
The answer is fairly clear: it'll be a bit like a bicategory, but instead of being able to compose $1$-cells straightforwardly, all we can do is specify, for any 'composable' sequence
$$
A_0 xrightarrow{f_1} cdots xrightarrow{f_n} A_n,,
$$
of $1$-cells and any $1$-cell $g colon A_0 to A_n$, the set of $2$-cells
$$
f_1,dots,f_n Rightarrow g,.
$$
What is the name of such a thing? A natural example, for a given SMCC $mathcal V$, has $mathcal V$-enriched categories as the objects, profunctors $mathcal C^{op}otimes mathcal Cto mathcal V$ as the $1$-cells and, for any collection
$$
mathcal C_0,cdots,mathcal C_n
$$
of categories and profunctors $F_icolon mathcal C_{i-1}^{op}otimes mathcal C_ito mathcal V$, $Gcolon mathcal C_0^{op}otimes mathcal C_n to mathcal V$, the set of extranatural transformations
$$
phi_{b_1,cdots,b_{n-1}} colon F_0(a,b_1) otimes cdots otimes F_n(b_{n-1},c)Rightarrow G(a,c)
$$
as the $2$-cells $F_1,cdots,F_n Rightarrow G$.
Of course, if $mathcal V$ is cocomplete, then this is equivalent to the usual bicategory of $mathcal V$-enriched profunctors.
reference-request ct.category-theory higher-category-theory enriched-category-theory profunctors
asked Jan 11 at 13:21
John GowersJohn Gowers
412213
$endgroup$
add a comment |
$begingroup$
We all know that
A monoidal category is a bicategory with one object.
How do we fill in the blank in the following sentence?
A multicategory is a ... with one object.
The answer is fairly clear: it'll be a bit like a bicategory, but instead of being able to compose $1$-cells straightforwardly, all we can do is specify, for any 'composable' sequence
$$
A_0 xrightarrow{f_1} cdots xrightarrow{f_n} A_n,,
$$
of $1$-cells and any $1$-cell $g colon A_0 to A_n$, the set of $2$-cells
$$
f_1,dots,f_n Rightarrow g,.
$$
What is the name of such a thing? A natural example, for a given SMCC $mathcal V$, has $mathcal V$-enriched categories as the objects, profunctors $mathcal C^{op}otimes mathcal Cto mathcal V$ as the $1$-cells and, for any collection
$$
mathcal C_0,cdots,mathcal C_n
$$
of categories and profunctors $F_icolon mathcal C_{i-1}^{op}otimes mathcal C_ito mathcal V$, $Gcolon mathcal C_0^{op}otimes mathcal C_n to mathcal V$, the set of extranatural transformations
$$
phi_{b_1,cdots,b_{n-1}} colon F_0(a,b_1) otimes cdots otimes F_n(b_{n-1},c)Rightarrow G(a,c)
$$
as the $2$-cells $F_1,cdots,F_n Rightarrow G$.
Of course, if $mathcal V$ is cocomplete, then this is equivalent to the usual bicategory of $mathcal V$-enriched profunctors.
reference-request ct.category-theory higher-category-theory enriched-category-theory profunctors
asked Jan 11 at 13:21
John GowersJohn Gowers
412213
$endgroup$
We all know that
A monoidal category is a bicategory with one object.
How do we fill in the blank in the following sentence?
A multicategory is a ... with one object.
The answer is fairly clear: it'll be a bit like a bicategory, but instead of being able to compose $1$-cells straightforwardly, all we can do is specify, for any 'composable' sequence
$$
A_0 xrightarrow{f_1} cdots xrightarrow{f_n} A_n,,
$$
of $1$-cells and any $1$-cell $g colon A_0 to A_n$, the set of $2$-cells
$$
f_1,dots,f_n Rightarrow g,.
$$
What is the name of such a thing? A natural example, for a given SMCC $mathcal V$, has $mathcal V$-enriched categories as the objects, profunctors $mathcal C^{op}otimes mathcal Cto mathcal V$ as the $1$-cells and, for any collection
$$
mathcal C_0,cdots,mathcal C_n
$$
of categories and profunctors $F_icolon mathcal C_{i-1}^{op}otimes mathcal C_ito mathcal V$, $Gcolon mathcal C_0^{op}otimes mathcal C_n to mathcal V$, the set of extranatural transformations
$$
phi_{b_1,cdots,b_{n-1}} colon F_0(a,b_1) otimes cdots otimes F_n(b_{n-1},c)Rightarrow G(a,c)
$$
as the $2$-cells $F_1,cdots,F_n Rightarrow G$.
Of course, if $mathcal V$ is cocomplete, then this is equivalent to the usual bicategory of $mathcal V$-enriched profunctors.
reference-request ct.category-theory higher-category-theory enriched-category-theory profunctors
reference-request ct.category-theory higher-category-theory enriched-category-theory profunctors
asked Jan 11 at 13:21
John GowersJohn Gowers
412213
asked Jan 11 at 13:21
John GowersJohn Gowers
412213
edited Jan 11 at 18:44
John Gowers
asked Jan 11 at 13:21
John GowersJohn Gowers
412213
asked Jan 11 at 13:21
John GowersJohn Gowers
412213
asked Jan 11 at 13:21
John GowersJohn Gowers
412213
412213
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
This has been called a "fc-multicategory" by Tom Leinster, for example here.
I think this as also been called a "Hypervirtual double category" here, but I don't remember if this is exactly the same notion or if there are some additional assumption in the second link.
answered Jan 11 at 13:47
Simon HenrySimon Henry
14.8k14885
$endgroup$
3
$begingroup$
fc-multicategories seem to generalize what I'm talking about slightly, by allowing vertical $1$-cells between objects as well. I suppose that that is a natural generalization: for example, in the profunctor case we can take the vertical $1$-cells to be the ordinary functors.
$endgroup$
– John Gowers
Jan 11 at 13:56
4
$begingroup$
Oh, yes you are write I missed that: the structure you are considering is a fc-multicategory with only identity vertical 1-cells.
$endgroup$
– Simon Henry
Jan 11 at 13:59
5
$begingroup$
Shulman and Cruttwell have been using the term "virtual double categories" for fc-multicategories. Hypervirtual double categories generalise these by also including cells with empty horizontal targets (besides the usual target: a single horizontal morphism). It looks like that, when restricting to a single identity as the set of vertical morphisms, a hypervirtual double category consists of a multicategory C equipped with a module C -|-> 1, in the sense of 2.3.6 of link, where 1 is the terminal multicategory.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:17
$begingroup$
This is interesting, it makes me wonder if there is a similar characterisation of general hypervirtual double categories.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:19
2
$begingroup$
Here's the paper where Cruttwell and Shulman introduced the "virtual" terminology. "A unified framework for generalized multicategories"
$endgroup$
– Tim Campion
Jan 11 at 17:13
add a comment |
$begingroup$
Another established structure very close to what you want is opetopic bicategories, which are equivalent to classical bicategories but formulated opetopically; see e.g. §3 of Cheng 2003, Opetopic bicategories: comparison with the classical theory, and the subsection Non-algebraic notions of bicategory at the end of §3.4 of Leinster 2003, Higher operads, higher categories.
Concretely, in an opetopic bicategory $B$, you have a graph of 0-cells and 1-cells like in a normal bicategory; the source of a 2-cell is not just a 1-cell but a composable string of 1-cells; 2-cell composition looks just like what you’d expect; and the 1-cell composition condition says that for every composable sequence of 1-cells, there’s a universal 2-cell out of them, for a certain sense of universality.
Keeping all of this except the last condition — call such a thing an opetopic bicategories minus 1-cell composition — seems to give exactly what you’re asking for. In the one-object case, the 1-cell composition condition is exactly what Leinster calls representability of a multicategory (Def 3.3.1, ibid.) — so adding this back recovers the equivalence between monoidal categories and one-object bicategories:
(monoidal category) = (multicategory with representability) = (one-object opetopic bicategory minus 1-cell composition, with 1-cell composition) = (one-object opetopic bicategory) = (one-object bicategory)
Comparing this with Simon Henry’s answer, I would expect (opetopic bicategories minus 1-cell composition) should be fairly concretely equivalent to (fc-multicategories with only identity vertical 1-cells); indeed, Leinster hints at such a connection in the subsection mentioned above, though he doesn’t spell it out precisely.
answered Jan 11 at 23:11
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,39713767
$endgroup$
add a comment |
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2 Answers
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oldest
votes
2 Answers
2
active
oldest
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$begingroup$
This has been called a "fc-multicategory" by Tom Leinster, for example here.
I think this as also been called a "Hypervirtual double category" here, but I don't remember if this is exactly the same notion or if there are some additional assumption in the second link.
answered Jan 11 at 13:47
Simon HenrySimon Henry
14.8k14885
$endgroup$
3
$begingroup$
fc-multicategories seem to generalize what I'm talking about slightly, by allowing vertical $1$-cells between objects as well. I suppose that that is a natural generalization: for example, in the profunctor case we can take the vertical $1$-cells to be the ordinary functors.
$endgroup$
– John Gowers
Jan 11 at 13:56
4
$begingroup$
Oh, yes you are write I missed that: the structure you are considering is a fc-multicategory with only identity vertical 1-cells.
$endgroup$
– Simon Henry
Jan 11 at 13:59
5
$begingroup$
Shulman and Cruttwell have been using the term "virtual double categories" for fc-multicategories. Hypervirtual double categories generalise these by also including cells with empty horizontal targets (besides the usual target: a single horizontal morphism). It looks like that, when restricting to a single identity as the set of vertical morphisms, a hypervirtual double category consists of a multicategory C equipped with a module C -|-> 1, in the sense of 2.3.6 of link, where 1 is the terminal multicategory.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:17
$begingroup$
This is interesting, it makes me wonder if there is a similar characterisation of general hypervirtual double categories.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:19
2
$begingroup$
Here's the paper where Cruttwell and Shulman introduced the "virtual" terminology. "A unified framework for generalized multicategories"
$endgroup$
– Tim Campion
Jan 11 at 17:13
add a comment |
$begingroup$
This has been called a "fc-multicategory" by Tom Leinster, for example here.
I think this as also been called a "Hypervirtual double category" here, but I don't remember if this is exactly the same notion or if there are some additional assumption in the second link.
answered Jan 11 at 13:47
Simon HenrySimon Henry
14.8k14885
$endgroup$
3
$begingroup$
fc-multicategories seem to generalize what I'm talking about slightly, by allowing vertical $1$-cells between objects as well. I suppose that that is a natural generalization: for example, in the profunctor case we can take the vertical $1$-cells to be the ordinary functors.
$endgroup$
– John Gowers
Jan 11 at 13:56
4
$begingroup$
Oh, yes you are write I missed that: the structure you are considering is a fc-multicategory with only identity vertical 1-cells.
$endgroup$
– Simon Henry
Jan 11 at 13:59
5
$begingroup$
Shulman and Cruttwell have been using the term "virtual double categories" for fc-multicategories. Hypervirtual double categories generalise these by also including cells with empty horizontal targets (besides the usual target: a single horizontal morphism). It looks like that, when restricting to a single identity as the set of vertical morphisms, a hypervirtual double category consists of a multicategory C equipped with a module C -|-> 1, in the sense of 2.3.6 of link, where 1 is the terminal multicategory.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:17
$begingroup$
This is interesting, it makes me wonder if there is a similar characterisation of general hypervirtual double categories.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:19
2
$begingroup$
Here's the paper where Cruttwell and Shulman introduced the "virtual" terminology. "A unified framework for generalized multicategories"
$endgroup$
– Tim Campion
Jan 11 at 17:13
add a comment |
$begingroup$
This has been called a "fc-multicategory" by Tom Leinster, for example here.
I think this as also been called a "Hypervirtual double category" here, but I don't remember if this is exactly the same notion or if there are some additional assumption in the second link.
answered Jan 11 at 13:47
Simon HenrySimon Henry
14.8k14885
$endgroup$
This has been called a "fc-multicategory" by Tom Leinster, for example here.
I think this as also been called a "Hypervirtual double category" here, but I don't remember if this is exactly the same notion or if there are some additional assumption in the second link.
answered Jan 11 at 13:47
Simon HenrySimon Henry
14.8k14885
answered Jan 11 at 13:47
Simon HenrySimon Henry
14.8k14885
answered Jan 11 at 13:47
Simon HenrySimon Henry
14.8k14885
answered Jan 11 at 13:47
Simon HenrySimon Henry
14.8k14885
14.8k14885
3
$begingroup$
fc-multicategories seem to generalize what I'm talking about slightly, by allowing vertical $1$-cells between objects as well. I suppose that that is a natural generalization: for example, in the profunctor case we can take the vertical $1$-cells to be the ordinary functors.
$endgroup$
– John Gowers
Jan 11 at 13:56
4
$begingroup$
Oh, yes you are write I missed that: the structure you are considering is a fc-multicategory with only identity vertical 1-cells.
$endgroup$
– Simon Henry
Jan 11 at 13:59
5
$begingroup$
Shulman and Cruttwell have been using the term "virtual double categories" for fc-multicategories. Hypervirtual double categories generalise these by also including cells with empty horizontal targets (besides the usual target: a single horizontal morphism). It looks like that, when restricting to a single identity as the set of vertical morphisms, a hypervirtual double category consists of a multicategory C equipped with a module C -|-> 1, in the sense of 2.3.6 of link, where 1 is the terminal multicategory.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:17
$begingroup$
This is interesting, it makes me wonder if there is a similar characterisation of general hypervirtual double categories.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:19
2
$begingroup$
Here's the paper where Cruttwell and Shulman introduced the "virtual" terminology. "A unified framework for generalized multicategories"
$endgroup$
– Tim Campion
Jan 11 at 17:13
add a comment |
3
$begingroup$
fc-multicategories seem to generalize what I'm talking about slightly, by allowing vertical $1$-cells between objects as well. I suppose that that is a natural generalization: for example, in the profunctor case we can take the vertical $1$-cells to be the ordinary functors.
$endgroup$
– John Gowers
Jan 11 at 13:56
4
$begingroup$
Oh, yes you are write I missed that: the structure you are considering is a fc-multicategory with only identity vertical 1-cells.
$endgroup$
– Simon Henry
Jan 11 at 13:59
5
$begingroup$
Shulman and Cruttwell have been using the term "virtual double categories" for fc-multicategories. Hypervirtual double categories generalise these by also including cells with empty horizontal targets (besides the usual target: a single horizontal morphism). It looks like that, when restricting to a single identity as the set of vertical morphisms, a hypervirtual double category consists of a multicategory C equipped with a module C -|-> 1, in the sense of 2.3.6 of link, where 1 is the terminal multicategory.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:17
$begingroup$
This is interesting, it makes me wonder if there is a similar characterisation of general hypervirtual double categories.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:19
2
$begingroup$
Here's the paper where Cruttwell and Shulman introduced the "virtual" terminology. "A unified framework for generalized multicategories"
$endgroup$
– Tim Campion
Jan 11 at 17:13
3
3
$begingroup$
fc-multicategories seem to generalize what I'm talking about slightly, by allowing vertical $1$-cells between objects as well. I suppose that that is a natural generalization: for example, in the profunctor case we can take the vertical $1$-cells to be the ordinary functors.
$endgroup$
– John Gowers
Jan 11 at 13:56
$begingroup$
fc-multicategories seem to generalize what I'm talking about slightly, by allowing vertical $1$-cells between objects as well. I suppose that that is a natural generalization: for example, in the profunctor case we can take the vertical $1$-cells to be the ordinary functors.
$endgroup$
– John Gowers
Jan 11 at 13:56
4
4
$begingroup$
Oh, yes you are write I missed that: the structure you are considering is a fc-multicategory with only identity vertical 1-cells.
$endgroup$
– Simon Henry
Jan 11 at 13:59
$begingroup$
Oh, yes you are write I missed that: the structure you are considering is a fc-multicategory with only identity vertical 1-cells.
$endgroup$
– Simon Henry
Jan 11 at 13:59
5
5
$begingroup$
Shulman and Cruttwell have been using the term "virtual double categories" for fc-multicategories. Hypervirtual double categories generalise these by also including cells with empty horizontal targets (besides the usual target: a single horizontal morphism). It looks like that, when restricting to a single identity as the set of vertical morphisms, a hypervirtual double category consists of a multicategory C equipped with a module C -|-> 1, in the sense of 2.3.6 of link, where 1 is the terminal multicategory.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:17
$begingroup$
Shulman and Cruttwell have been using the term "virtual double categories" for fc-multicategories. Hypervirtual double categories generalise these by also including cells with empty horizontal targets (besides the usual target: a single horizontal morphism). It looks like that, when restricting to a single identity as the set of vertical morphisms, a hypervirtual double category consists of a multicategory C equipped with a module C -|-> 1, in the sense of 2.3.6 of link, where 1 is the terminal multicategory.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:17
$begingroup$
This is interesting, it makes me wonder if there is a similar characterisation of general hypervirtual double categories.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:19
$begingroup$
This is interesting, it makes me wonder if there is a similar characterisation of general hypervirtual double categories.
$endgroup$
– Roald Koudenburg
Jan 11 at 16:19
2
2
$begingroup$
Here's the paper where Cruttwell and Shulman introduced the "virtual" terminology. "A unified framework for generalized multicategories"
$endgroup$
– Tim Campion
Jan 11 at 17:13
$begingroup$
Here's the paper where Cruttwell and Shulman introduced the "virtual" terminology. "A unified framework for generalized multicategories"
$endgroup$
– Tim Campion
Jan 11 at 17:13
add a comment |
$begingroup$
Another established structure very close to what you want is opetopic bicategories, which are equivalent to classical bicategories but formulated opetopically; see e.g. §3 of Cheng 2003, Opetopic bicategories: comparison with the classical theory, and the subsection Non-algebraic notions of bicategory at the end of §3.4 of Leinster 2003, Higher operads, higher categories.
Concretely, in an opetopic bicategory $B$, you have a graph of 0-cells and 1-cells like in a normal bicategory; the source of a 2-cell is not just a 1-cell but a composable string of 1-cells; 2-cell composition looks just like what you’d expect; and the 1-cell composition condition says that for every composable sequence of 1-cells, there’s a universal 2-cell out of them, for a certain sense of universality.
Keeping all of this except the last condition — call such a thing an opetopic bicategories minus 1-cell composition — seems to give exactly what you’re asking for. In the one-object case, the 1-cell composition condition is exactly what Leinster calls representability of a multicategory (Def 3.3.1, ibid.) — so adding this back recovers the equivalence between monoidal categories and one-object bicategories:
(monoidal category) = (multicategory with representability) = (one-object opetopic bicategory minus 1-cell composition, with 1-cell composition) = (one-object opetopic bicategory) = (one-object bicategory)
Comparing this with Simon Henry’s answer, I would expect (opetopic bicategories minus 1-cell composition) should be fairly concretely equivalent to (fc-multicategories with only identity vertical 1-cells); indeed, Leinster hints at such a connection in the subsection mentioned above, though he doesn’t spell it out precisely.
answered Jan 11 at 23:11
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,39713767
$endgroup$
add a comment |
$begingroup$
Another established structure very close to what you want is opetopic bicategories, which are equivalent to classical bicategories but formulated opetopically; see e.g. §3 of Cheng 2003, Opetopic bicategories: comparison with the classical theory, and the subsection Non-algebraic notions of bicategory at the end of §3.4 of Leinster 2003, Higher operads, higher categories.
Concretely, in an opetopic bicategory $B$, you have a graph of 0-cells and 1-cells like in a normal bicategory; the source of a 2-cell is not just a 1-cell but a composable string of 1-cells; 2-cell composition looks just like what you’d expect; and the 1-cell composition condition says that for every composable sequence of 1-cells, there’s a universal 2-cell out of them, for a certain sense of universality.
Keeping all of this except the last condition — call such a thing an opetopic bicategories minus 1-cell composition — seems to give exactly what you’re asking for. In the one-object case, the 1-cell composition condition is exactly what Leinster calls representability of a multicategory (Def 3.3.1, ibid.) — so adding this back recovers the equivalence between monoidal categories and one-object bicategories:
(monoidal category) = (multicategory with representability) = (one-object opetopic bicategory minus 1-cell composition, with 1-cell composition) = (one-object opetopic bicategory) = (one-object bicategory)
Comparing this with Simon Henry’s answer, I would expect (opetopic bicategories minus 1-cell composition) should be fairly concretely equivalent to (fc-multicategories with only identity vertical 1-cells); indeed, Leinster hints at such a connection in the subsection mentioned above, though he doesn’t spell it out precisely.
answered Jan 11 at 23:11
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,39713767
$endgroup$
add a comment |
$begingroup$
Another established structure very close to what you want is opetopic bicategories, which are equivalent to classical bicategories but formulated opetopically; see e.g. §3 of Cheng 2003, Opetopic bicategories: comparison with the classical theory, and the subsection Non-algebraic notions of bicategory at the end of §3.4 of Leinster 2003, Higher operads, higher categories.
Concretely, in an opetopic bicategory $B$, you have a graph of 0-cells and 1-cells like in a normal bicategory; the source of a 2-cell is not just a 1-cell but a composable string of 1-cells; 2-cell composition looks just like what you’d expect; and the 1-cell composition condition says that for every composable sequence of 1-cells, there’s a universal 2-cell out of them, for a certain sense of universality.
Keeping all of this except the last condition — call such a thing an opetopic bicategories minus 1-cell composition — seems to give exactly what you’re asking for. In the one-object case, the 1-cell composition condition is exactly what Leinster calls representability of a multicategory (Def 3.3.1, ibid.) — so adding this back recovers the equivalence between monoidal categories and one-object bicategories:
(monoidal category) = (multicategory with representability) = (one-object opetopic bicategory minus 1-cell composition, with 1-cell composition) = (one-object opetopic bicategory) = (one-object bicategory)
Comparing this with Simon Henry’s answer, I would expect (opetopic bicategories minus 1-cell composition) should be fairly concretely equivalent to (fc-multicategories with only identity vertical 1-cells); indeed, Leinster hints at such a connection in the subsection mentioned above, though he doesn’t spell it out precisely.
answered Jan 11 at 23:11
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,39713767
$endgroup$
Another established structure very close to what you want is opetopic bicategories, which are equivalent to classical bicategories but formulated opetopically; see e.g. §3 of Cheng 2003, Opetopic bicategories: comparison with the classical theory, and the subsection Non-algebraic notions of bicategory at the end of §3.4 of Leinster 2003, Higher operads, higher categories.
Concretely, in an opetopic bicategory $B$, you have a graph of 0-cells and 1-cells like in a normal bicategory; the source of a 2-cell is not just a 1-cell but a composable string of 1-cells; 2-cell composition looks just like what you’d expect; and the 1-cell composition condition says that for every composable sequence of 1-cells, there’s a universal 2-cell out of them, for a certain sense of universality.
Keeping all of this except the last condition — call such a thing an opetopic bicategories minus 1-cell composition — seems to give exactly what you’re asking for. In the one-object case, the 1-cell composition condition is exactly what Leinster calls representability of a multicategory (Def 3.3.1, ibid.) — so adding this back recovers the equivalence between monoidal categories and one-object bicategories:
(monoidal category) = (multicategory with representability) = (one-object opetopic bicategory minus 1-cell composition, with 1-cell composition) = (one-object opetopic bicategory) = (one-object bicategory)
Comparing this with Simon Henry’s answer, I would expect (opetopic bicategories minus 1-cell composition) should be fairly concretely equivalent to (fc-multicategories with only identity vertical 1-cells); indeed, Leinster hints at such a connection in the subsection mentioned above, though he doesn’t spell it out precisely.
answered Jan 11 at 23:11
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,39713767
answered Jan 11 at 23:11
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,39713767
answered Jan 11 at 23:11
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,39713767
answered Jan 11 at 23:11
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,39713767
8,39713767
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