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what is an ∞-group?
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I was reading on nLab and I found the term infinity group. The definition is awfully abstract:
An ∞-group is a group object in ∞Grpd.
Equivalently (by the delooping hypothesis) it is a pointed connected ∞-groupoid.
Under the identification of ∞Grpd with Top this is known as an A∞-space, for instance.
Then I tried to look up the term "group-object" and obtained:
A group object in an (∞,1)-category generalizes and unifies two familiar concepts:
it is the generalization of the notion of groupal Stasheff A∞-space from Top to more general (∞,1)-sheaf (∞,1)-toposes: an object that comes equipped with an associative and invertible monoid structure, up to coherent homotopy, and possibly only partially defined (see also looping and delooping for more on this) ;
it generalizes the notion of equivalence relation – or rather the internal notion of congruence – from category theory to (∞,1)-category theory.
At this point I don't even know enough to specify what an answer should look like. As I recall, a group $G$ is a set with a binary operation $times$ so that $(G,times)$ should have three properties:
- an identity element $e in G$ with $e times g = g times e = g$
- every element $g in G$ has an inverse $g^-1$ so that $g times g^-1= g^-1 times g = e$
- multiplication is associative $a times (b times c) = (a times b) times c$ for all $a,b,c in G$.
So our infinity group looks nothing like the object that i am familiar with.
group-theory category-theory homotopy-theory groupoids
asked 4 hours ago
cactus314
15.1k41863
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up vote
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I was reading on nLab and I found the term infinity group. The definition is awfully abstract:
An ∞-group is a group object in ∞Grpd.
Equivalently (by the delooping hypothesis) it is a pointed connected ∞-groupoid.
Under the identification of ∞Grpd with Top this is known as an A∞-space, for instance.
Then I tried to look up the term "group-object" and obtained:
A group object in an (∞,1)-category generalizes and unifies two familiar concepts:
it is the generalization of the notion of groupal Stasheff A∞-space from Top to more general (∞,1)-sheaf (∞,1)-toposes: an object that comes equipped with an associative and invertible monoid structure, up to coherent homotopy, and possibly only partially defined (see also looping and delooping for more on this) ;
it generalizes the notion of equivalence relation – or rather the internal notion of congruence – from category theory to (∞,1)-category theory.
At this point I don't even know enough to specify what an answer should look like. As I recall, a group $G$ is a set with a binary operation $times$ so that $(G,times)$ should have three properties:
- an identity element $e in G$ with $e times g = g times e = g$
- every element $g in G$ has an inverse $g^-1$ so that $g times g^-1= g^-1 times g = e$
- multiplication is associative $a times (b times c) = (a times b) times c$ for all $a,b,c in G$.
So our infinity group looks nothing like the object that i am familiar with.
group-theory category-theory homotopy-theory groupoids
asked 4 hours ago
cactus314
15.1k41863
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
I was reading on nLab and I found the term infinity group. The definition is awfully abstract:
An ∞-group is a group object in ∞Grpd.
Equivalently (by the delooping hypothesis) it is a pointed connected ∞-groupoid.
Under the identification of ∞Grpd with Top this is known as an A∞-space, for instance.
Then I tried to look up the term "group-object" and obtained:
A group object in an (∞,1)-category generalizes and unifies two familiar concepts:
it is the generalization of the notion of groupal Stasheff A∞-space from Top to more general (∞,1)-sheaf (∞,1)-toposes: an object that comes equipped with an associative and invertible monoid structure, up to coherent homotopy, and possibly only partially defined (see also looping and delooping for more on this) ;
it generalizes the notion of equivalence relation – or rather the internal notion of congruence – from category theory to (∞,1)-category theory.
At this point I don't even know enough to specify what an answer should look like. As I recall, a group $G$ is a set with a binary operation $times$ so that $(G,times)$ should have three properties:
- an identity element $e in G$ with $e times g = g times e = g$
- every element $g in G$ has an inverse $g^-1$ so that $g times g^-1= g^-1 times g = e$
- multiplication is associative $a times (b times c) = (a times b) times c$ for all $a,b,c in G$.
So our infinity group looks nothing like the object that i am familiar with.
group-theory category-theory homotopy-theory groupoids
asked 4 hours ago
cactus314
15.1k41863
I was reading on nLab and I found the term infinity group. The definition is awfully abstract:
An ∞-group is a group object in ∞Grpd.
Equivalently (by the delooping hypothesis) it is a pointed connected ∞-groupoid.
Under the identification of ∞Grpd with Top this is known as an A∞-space, for instance.
Then I tried to look up the term "group-object" and obtained:
A group object in an (∞,1)-category generalizes and unifies two familiar concepts:
it is the generalization of the notion of groupal Stasheff A∞-space from Top to more general (∞,1)-sheaf (∞,1)-toposes: an object that comes equipped with an associative and invertible monoid structure, up to coherent homotopy, and possibly only partially defined (see also looping and delooping for more on this) ;
it generalizes the notion of equivalence relation – or rather the internal notion of congruence – from category theory to (∞,1)-category theory.
At this point I don't even know enough to specify what an answer should look like. As I recall, a group $G$ is a set with a binary operation $times$ so that $(G,times)$ should have three properties:
- an identity element $e in G$ with $e times g = g times e = g$
- every element $g in G$ has an inverse $g^-1$ so that $g times g^-1= g^-1 times g = e$
- multiplication is associative $a times (b times c) = (a times b) times c$ for all $a,b,c in G$.
So our infinity group looks nothing like the object that i am familiar with.
group-theory category-theory homotopy-theory groupoids
group-theory category-theory homotopy-theory groupoids
asked 4 hours ago
cactus314
15.1k41863
asked 4 hours ago
cactus314
15.1k41863
asked 4 hours ago
cactus314
15.1k41863
asked 4 hours ago
cactus314
15.1k41863
asked 4 hours ago
cactus314
15.1k41863
15.1k41863
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At a certain level, an $infty$-group $G$ is not so different from the groups you are used to. $G$ has an underlying space, analogous to the underlying set of a group, a unit $e:*to G$, a multiplication $m:Gtimes Gto G$, and an inversion map $i:Gto G$. Where things begin to get interesting is the sense in which these operations satisfy the group axioms. It is no longer true that $m(m(g_1,g_2),g_3)=m(g_1,m(g_2,g_3))$; rather, this associativity axiom becomes yet another piece of structure, an associator map $a:Gtimes Gtimes Gtimes Ito G$ giving a homotopy between the two sides of the above equation. Similarly, we have to introduce "unitors" replacing the equations $m(g,e)=g$ and $m(e,g)=g$ with homotopies, and similarly for the equations involving the inverse. Furthermore, this isn't all! In fact a group satisfies many more equations than those in the usual axiomatization. For instance, there are five different ways to parentheses four letters: $g_1(g_2(g_3g_4)),(g_1g_2)(g_3g_4),((g_1g_2)g_3)g_4),(g_1(g_2g_3))g_4,$ and $g_1((g_2g_3)g_4)$. In a group, these are all equal, and this follows from the associativity axiom. In an $infty$-group, we have homotopies between these five parenthesizations, interpreted as maps $G^4to G$. In fact, we can paste these homotopies together into a map $G^4times partial Pto G$, where $P$ is a regular pentagon in the plane. We would like to know that there is, in essence, only one way to associate two products into each other-it would be a bad generalization of group theory if we could follow a nontrivial loop in $G$ by simply associating one word back to itself in some complicated sequence! Thus part of the structure of an $infty$-group is an extension of the above map to the pentagonator $pi:G^4times Pto G$.
And we're not done yet. In fact, there are infinitely many levels of structure needed for describing all the ways of associating longer and longer words alone, and the spaces which are $I$ for the associator and $P$ for the pentagonator continue growing in dimension and combinatorial complexity. Stasheff gave the first complete description of this part of the structure of an $infty$-group, which he called an $A_infty$-space, for a space with a multiplication which is Associative up to a homotopy which is itself well defined up to a homotopy which is well defined up to...Stasheff's original papers are still excellent reading on this topic.
The nLab's "group object" in an $infty$-category is closely related to Stasheff's notion of $A_infty$-space and in similar ways to the elementary notion of a group. The "simplical object in an $infty$-category" that is the underlying structure of a group object is supposed to represent the group $G$ together with all its finite powers $G^n$ (including $n=0$) while the simplicial face maps correspond to the canonical projections between $G^n$ and $G^m$, the degeneracies correspond to various ways of mapping $G^m$ to $G^n$ by inserting copies of the unit, and the various pullback squares cleverly encode the multiplication, inverse, and all the infinite tower of homotopies witnessing the axioms as in the previous paragraph. This is also closely connected to Lawvere's perspective on groups: a model of the Lawvere theory of groups in a category $C$ is exactly all the stuff I just said, except that the homotopies are allowed again to be equations. So it's no different than an ordinary group object, except insofar as we don't pick out particular operations and axioms as privileged.
This is a pretty complicated structure! A large amount of work in the last fifty years of algebraic topology has been on how best to understand these objects. One fundamental theorem is that an $A_infty$-space has a delooping if and only if it is actually an $infty$-group: having a delooping means it is homotopy equivalent to the space of based loops in some pointed connected space, which is itself unique up to homotopy equivalence. And a map of $infty$-groups, i.e. some kind of homomorphism appropriately preserving all the huge mess of structure up to a huge number of homotopies, is nothing more than a map of their deloopings. (This statement is a little bit cleaner than the reality, but it's close.) This is the equivalence between $infty$-groups and pointed connected objects you mention. This has less of an obvious analogue to ordinary group theory, but it's still there: it's simply the well-known perverse definition of a group as a groupoid with a single object. The reason the equivalence is so much more interesting in $infty$-category theory is that pointed connected $infty$-groupoids, i.e. pointed connected spaces, are usually not defined with algebraic operations of composition and inversion of loops, so much less structure has to be carried around in defining them and in particular their maps. Another simplification is that every $infty$-group is appropriately equivalent to the geometric realization of a simplicial group, that is, a completely ordinary group object in the ordinary category of simplicial sets. At least the equivalence between pointed connected spaces and simplicial groups is the oldest of all these results-it goes back to Kan in the early '60s.
Anyway, hopefully that's given a bit more of an idea of what's going on. There are many approaches to the concept, largely because all of the approaches become intolerably complicated in one way or another. This situation is characteristic of $infty$-category theory, and in the current state of knowledge it appears unavoidable that appending "$infty$" to a familiar object creates substantial complications.
answered 1 hour ago
Kevin Carlson
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At a certain level, an $infty$-group $G$ is not so different from the groups you are used to. $G$ has an underlying space, analogous to the underlying set of a group, a unit $e:*to G$, a multiplication $m:Gtimes Gto G$, and an inversion map $i:Gto G$. Where things begin to get interesting is the sense in which these operations satisfy the group axioms. It is no longer true that $m(m(g_1,g_2),g_3)=m(g_1,m(g_2,g_3))$; rather, this associativity axiom becomes yet another piece of structure, an associator map $a:Gtimes Gtimes Gtimes Ito G$ giving a homotopy between the two sides of the above equation. Similarly, we have to introduce "unitors" replacing the equations $m(g,e)=g$ and $m(e,g)=g$ with homotopies, and similarly for the equations involving the inverse. Furthermore, this isn't all! In fact a group satisfies many more equations than those in the usual axiomatization. For instance, there are five different ways to parentheses four letters: $g_1(g_2(g_3g_4)),(g_1g_2)(g_3g_4),((g_1g_2)g_3)g_4),(g_1(g_2g_3))g_4,$ and $g_1((g_2g_3)g_4)$. In a group, these are all equal, and this follows from the associativity axiom. In an $infty$-group, we have homotopies between these five parenthesizations, interpreted as maps $G^4to G$. In fact, we can paste these homotopies together into a map $G^4times partial Pto G$, where $P$ is a regular pentagon in the plane. We would like to know that there is, in essence, only one way to associate two products into each other-it would be a bad generalization of group theory if we could follow a nontrivial loop in $G$ by simply associating one word back to itself in some complicated sequence! Thus part of the structure of an $infty$-group is an extension of the above map to the pentagonator $pi:G^4times Pto G$.
And we're not done yet. In fact, there are infinitely many levels of structure needed for describing all the ways of associating longer and longer words alone, and the spaces which are $I$ for the associator and $P$ for the pentagonator continue growing in dimension and combinatorial complexity. Stasheff gave the first complete description of this part of the structure of an $infty$-group, which he called an $A_infty$-space, for a space with a multiplication which is Associative up to a homotopy which is itself well defined up to a homotopy which is well defined up to...Stasheff's original papers are still excellent reading on this topic.
The nLab's "group object" in an $infty$-category is closely related to Stasheff's notion of $A_infty$-space and in similar ways to the elementary notion of a group. The "simplical object in an $infty$-category" that is the underlying structure of a group object is supposed to represent the group $G$ together with all its finite powers $G^n$ (including $n=0$) while the simplicial face maps correspond to the canonical projections between $G^n$ and $G^m$, the degeneracies correspond to various ways of mapping $G^m$ to $G^n$ by inserting copies of the unit, and the various pullback squares cleverly encode the multiplication, inverse, and all the infinite tower of homotopies witnessing the axioms as in the previous paragraph. This is also closely connected to Lawvere's perspective on groups: a model of the Lawvere theory of groups in a category $C$ is exactly all the stuff I just said, except that the homotopies are allowed again to be equations. So it's no different than an ordinary group object, except insofar as we don't pick out particular operations and axioms as privileged.
This is a pretty complicated structure! A large amount of work in the last fifty years of algebraic topology has been on how best to understand these objects. One fundamental theorem is that an $A_infty$-space has a delooping if and only if it is actually an $infty$-group: having a delooping means it is homotopy equivalent to the space of based loops in some pointed connected space, which is itself unique up to homotopy equivalence. And a map of $infty$-groups, i.e. some kind of homomorphism appropriately preserving all the huge mess of structure up to a huge number of homotopies, is nothing more than a map of their deloopings. (This statement is a little bit cleaner than the reality, but it's close.) This is the equivalence between $infty$-groups and pointed connected objects you mention. This has less of an obvious analogue to ordinary group theory, but it's still there: it's simply the well-known perverse definition of a group as a groupoid with a single object. The reason the equivalence is so much more interesting in $infty$-category theory is that pointed connected $infty$-groupoids, i.e. pointed connected spaces, are usually not defined with algebraic operations of composition and inversion of loops, so much less structure has to be carried around in defining them and in particular their maps. Another simplification is that every $infty$-group is appropriately equivalent to the geometric realization of a simplicial group, that is, a completely ordinary group object in the ordinary category of simplicial sets. At least the equivalence between pointed connected spaces and simplicial groups is the oldest of all these results-it goes back to Kan in the early '60s.
Anyway, hopefully that's given a bit more of an idea of what's going on. There are many approaches to the concept, largely because all of the approaches become intolerably complicated in one way or another. This situation is characteristic of $infty$-category theory, and in the current state of knowledge it appears unavoidable that appending "$infty$" to a familiar object creates substantial complications.
answered 1 hour ago
Kevin Carlson
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up vote
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At a certain level, an $infty$-group $G$ is not so different from the groups you are used to. $G$ has an underlying space, analogous to the underlying set of a group, a unit $e:*to G$, a multiplication $m:Gtimes Gto G$, and an inversion map $i:Gto G$. Where things begin to get interesting is the sense in which these operations satisfy the group axioms. It is no longer true that $m(m(g_1,g_2),g_3)=m(g_1,m(g_2,g_3))$; rather, this associativity axiom becomes yet another piece of structure, an associator map $a:Gtimes Gtimes Gtimes Ito G$ giving a homotopy between the two sides of the above equation. Similarly, we have to introduce "unitors" replacing the equations $m(g,e)=g$ and $m(e,g)=g$ with homotopies, and similarly for the equations involving the inverse. Furthermore, this isn't all! In fact a group satisfies many more equations than those in the usual axiomatization. For instance, there are five different ways to parentheses four letters: $g_1(g_2(g_3g_4)),(g_1g_2)(g_3g_4),((g_1g_2)g_3)g_4),(g_1(g_2g_3))g_4,$ and $g_1((g_2g_3)g_4)$. In a group, these are all equal, and this follows from the associativity axiom. In an $infty$-group, we have homotopies between these five parenthesizations, interpreted as maps $G^4to G$. In fact, we can paste these homotopies together into a map $G^4times partial Pto G$, where $P$ is a regular pentagon in the plane. We would like to know that there is, in essence, only one way to associate two products into each other-it would be a bad generalization of group theory if we could follow a nontrivial loop in $G$ by simply associating one word back to itself in some complicated sequence! Thus part of the structure of an $infty$-group is an extension of the above map to the pentagonator $pi:G^4times Pto G$.
And we're not done yet. In fact, there are infinitely many levels of structure needed for describing all the ways of associating longer and longer words alone, and the spaces which are $I$ for the associator and $P$ for the pentagonator continue growing in dimension and combinatorial complexity. Stasheff gave the first complete description of this part of the structure of an $infty$-group, which he called an $A_infty$-space, for a space with a multiplication which is Associative up to a homotopy which is itself well defined up to a homotopy which is well defined up to...Stasheff's original papers are still excellent reading on this topic.
The nLab's "group object" in an $infty$-category is closely related to Stasheff's notion of $A_infty$-space and in similar ways to the elementary notion of a group. The "simplical object in an $infty$-category" that is the underlying structure of a group object is supposed to represent the group $G$ together with all its finite powers $G^n$ (including $n=0$) while the simplicial face maps correspond to the canonical projections between $G^n$ and $G^m$, the degeneracies correspond to various ways of mapping $G^m$ to $G^n$ by inserting copies of the unit, and the various pullback squares cleverly encode the multiplication, inverse, and all the infinite tower of homotopies witnessing the axioms as in the previous paragraph. This is also closely connected to Lawvere's perspective on groups: a model of the Lawvere theory of groups in a category $C$ is exactly all the stuff I just said, except that the homotopies are allowed again to be equations. So it's no different than an ordinary group object, except insofar as we don't pick out particular operations and axioms as privileged.
This is a pretty complicated structure! A large amount of work in the last fifty years of algebraic topology has been on how best to understand these objects. One fundamental theorem is that an $A_infty$-space has a delooping if and only if it is actually an $infty$-group: having a delooping means it is homotopy equivalent to the space of based loops in some pointed connected space, which is itself unique up to homotopy equivalence. And a map of $infty$-groups, i.e. some kind of homomorphism appropriately preserving all the huge mess of structure up to a huge number of homotopies, is nothing more than a map of their deloopings. (This statement is a little bit cleaner than the reality, but it's close.) This is the equivalence between $infty$-groups and pointed connected objects you mention. This has less of an obvious analogue to ordinary group theory, but it's still there: it's simply the well-known perverse definition of a group as a groupoid with a single object. The reason the equivalence is so much more interesting in $infty$-category theory is that pointed connected $infty$-groupoids, i.e. pointed connected spaces, are usually not defined with algebraic operations of composition and inversion of loops, so much less structure has to be carried around in defining them and in particular their maps. Another simplification is that every $infty$-group is appropriately equivalent to the geometric realization of a simplicial group, that is, a completely ordinary group object in the ordinary category of simplicial sets. At least the equivalence between pointed connected spaces and simplicial groups is the oldest of all these results-it goes back to Kan in the early '60s.
Anyway, hopefully that's given a bit more of an idea of what's going on. There are many approaches to the concept, largely because all of the approaches become intolerably complicated in one way or another. This situation is characteristic of $infty$-category theory, and in the current state of knowledge it appears unavoidable that appending "$infty$" to a familiar object creates substantial complications.
answered 1 hour ago
Kevin Carlson
31.4k23269
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up vote
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up vote
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down vote
At a certain level, an $infty$-group $G$ is not so different from the groups you are used to. $G$ has an underlying space, analogous to the underlying set of a group, a unit $e:*to G$, a multiplication $m:Gtimes Gto G$, and an inversion map $i:Gto G$. Where things begin to get interesting is the sense in which these operations satisfy the group axioms. It is no longer true that $m(m(g_1,g_2),g_3)=m(g_1,m(g_2,g_3))$; rather, this associativity axiom becomes yet another piece of structure, an associator map $a:Gtimes Gtimes Gtimes Ito G$ giving a homotopy between the two sides of the above equation. Similarly, we have to introduce "unitors" replacing the equations $m(g,e)=g$ and $m(e,g)=g$ with homotopies, and similarly for the equations involving the inverse. Furthermore, this isn't all! In fact a group satisfies many more equations than those in the usual axiomatization. For instance, there are five different ways to parentheses four letters: $g_1(g_2(g_3g_4)),(g_1g_2)(g_3g_4),((g_1g_2)g_3)g_4),(g_1(g_2g_3))g_4,$ and $g_1((g_2g_3)g_4)$. In a group, these are all equal, and this follows from the associativity axiom. In an $infty$-group, we have homotopies between these five parenthesizations, interpreted as maps $G^4to G$. In fact, we can paste these homotopies together into a map $G^4times partial Pto G$, where $P$ is a regular pentagon in the plane. We would like to know that there is, in essence, only one way to associate two products into each other-it would be a bad generalization of group theory if we could follow a nontrivial loop in $G$ by simply associating one word back to itself in some complicated sequence! Thus part of the structure of an $infty$-group is an extension of the above map to the pentagonator $pi:G^4times Pto G$.
And we're not done yet. In fact, there are infinitely many levels of structure needed for describing all the ways of associating longer and longer words alone, and the spaces which are $I$ for the associator and $P$ for the pentagonator continue growing in dimension and combinatorial complexity. Stasheff gave the first complete description of this part of the structure of an $infty$-group, which he called an $A_infty$-space, for a space with a multiplication which is Associative up to a homotopy which is itself well defined up to a homotopy which is well defined up to...Stasheff's original papers are still excellent reading on this topic.
The nLab's "group object" in an $infty$-category is closely related to Stasheff's notion of $A_infty$-space and in similar ways to the elementary notion of a group. The "simplical object in an $infty$-category" that is the underlying structure of a group object is supposed to represent the group $G$ together with all its finite powers $G^n$ (including $n=0$) while the simplicial face maps correspond to the canonical projections between $G^n$ and $G^m$, the degeneracies correspond to various ways of mapping $G^m$ to $G^n$ by inserting copies of the unit, and the various pullback squares cleverly encode the multiplication, inverse, and all the infinite tower of homotopies witnessing the axioms as in the previous paragraph. This is also closely connected to Lawvere's perspective on groups: a model of the Lawvere theory of groups in a category $C$ is exactly all the stuff I just said, except that the homotopies are allowed again to be equations. So it's no different than an ordinary group object, except insofar as we don't pick out particular operations and axioms as privileged.
This is a pretty complicated structure! A large amount of work in the last fifty years of algebraic topology has been on how best to understand these objects. One fundamental theorem is that an $A_infty$-space has a delooping if and only if it is actually an $infty$-group: having a delooping means it is homotopy equivalent to the space of based loops in some pointed connected space, which is itself unique up to homotopy equivalence. And a map of $infty$-groups, i.e. some kind of homomorphism appropriately preserving all the huge mess of structure up to a huge number of homotopies, is nothing more than a map of their deloopings. (This statement is a little bit cleaner than the reality, but it's close.) This is the equivalence between $infty$-groups and pointed connected objects you mention. This has less of an obvious analogue to ordinary group theory, but it's still there: it's simply the well-known perverse definition of a group as a groupoid with a single object. The reason the equivalence is so much more interesting in $infty$-category theory is that pointed connected $infty$-groupoids, i.e. pointed connected spaces, are usually not defined with algebraic operations of composition and inversion of loops, so much less structure has to be carried around in defining them and in particular their maps. Another simplification is that every $infty$-group is appropriately equivalent to the geometric realization of a simplicial group, that is, a completely ordinary group object in the ordinary category of simplicial sets. At least the equivalence between pointed connected spaces and simplicial groups is the oldest of all these results-it goes back to Kan in the early '60s.
Anyway, hopefully that's given a bit more of an idea of what's going on. There are many approaches to the concept, largely because all of the approaches become intolerably complicated in one way or another. This situation is characteristic of $infty$-category theory, and in the current state of knowledge it appears unavoidable that appending "$infty$" to a familiar object creates substantial complications.
answered 1 hour ago
Kevin Carlson
31.4k23269
At a certain level, an $infty$-group $G$ is not so different from the groups you are used to. $G$ has an underlying space, analogous to the underlying set of a group, a unit $e:*to G$, a multiplication $m:Gtimes Gto G$, and an inversion map $i:Gto G$. Where things begin to get interesting is the sense in which these operations satisfy the group axioms. It is no longer true that $m(m(g_1,g_2),g_3)=m(g_1,m(g_2,g_3))$; rather, this associativity axiom becomes yet another piece of structure, an associator map $a:Gtimes Gtimes Gtimes Ito G$ giving a homotopy between the two sides of the above equation. Similarly, we have to introduce "unitors" replacing the equations $m(g,e)=g$ and $m(e,g)=g$ with homotopies, and similarly for the equations involving the inverse. Furthermore, this isn't all! In fact a group satisfies many more equations than those in the usual axiomatization. For instance, there are five different ways to parentheses four letters: $g_1(g_2(g_3g_4)),(g_1g_2)(g_3g_4),((g_1g_2)g_3)g_4),(g_1(g_2g_3))g_4,$ and $g_1((g_2g_3)g_4)$. In a group, these are all equal, and this follows from the associativity axiom. In an $infty$-group, we have homotopies between these five parenthesizations, interpreted as maps $G^4to G$. In fact, we can paste these homotopies together into a map $G^4times partial Pto G$, where $P$ is a regular pentagon in the plane. We would like to know that there is, in essence, only one way to associate two products into each other-it would be a bad generalization of group theory if we could follow a nontrivial loop in $G$ by simply associating one word back to itself in some complicated sequence! Thus part of the structure of an $infty$-group is an extension of the above map to the pentagonator $pi:G^4times Pto G$.
And we're not done yet. In fact, there are infinitely many levels of structure needed for describing all the ways of associating longer and longer words alone, and the spaces which are $I$ for the associator and $P$ for the pentagonator continue growing in dimension and combinatorial complexity. Stasheff gave the first complete description of this part of the structure of an $infty$-group, which he called an $A_infty$-space, for a space with a multiplication which is Associative up to a homotopy which is itself well defined up to a homotopy which is well defined up to...Stasheff's original papers are still excellent reading on this topic.
The nLab's "group object" in an $infty$-category is closely related to Stasheff's notion of $A_infty$-space and in similar ways to the elementary notion of a group. The "simplical object in an $infty$-category" that is the underlying structure of a group object is supposed to represent the group $G$ together with all its finite powers $G^n$ (including $n=0$) while the simplicial face maps correspond to the canonical projections between $G^n$ and $G^m$, the degeneracies correspond to various ways of mapping $G^m$ to $G^n$ by inserting copies of the unit, and the various pullback squares cleverly encode the multiplication, inverse, and all the infinite tower of homotopies witnessing the axioms as in the previous paragraph. This is also closely connected to Lawvere's perspective on groups: a model of the Lawvere theory of groups in a category $C$ is exactly all the stuff I just said, except that the homotopies are allowed again to be equations. So it's no different than an ordinary group object, except insofar as we don't pick out particular operations and axioms as privileged.
This is a pretty complicated structure! A large amount of work in the last fifty years of algebraic topology has been on how best to understand these objects. One fundamental theorem is that an $A_infty$-space has a delooping if and only if it is actually an $infty$-group: having a delooping means it is homotopy equivalent to the space of based loops in some pointed connected space, which is itself unique up to homotopy equivalence. And a map of $infty$-groups, i.e. some kind of homomorphism appropriately preserving all the huge mess of structure up to a huge number of homotopies, is nothing more than a map of their deloopings. (This statement is a little bit cleaner than the reality, but it's close.) This is the equivalence between $infty$-groups and pointed connected objects you mention. This has less of an obvious analogue to ordinary group theory, but it's still there: it's simply the well-known perverse definition of a group as a groupoid with a single object. The reason the equivalence is so much more interesting in $infty$-category theory is that pointed connected $infty$-groupoids, i.e. pointed connected spaces, are usually not defined with algebraic operations of composition and inversion of loops, so much less structure has to be carried around in defining them and in particular their maps. Another simplification is that every $infty$-group is appropriately equivalent to the geometric realization of a simplicial group, that is, a completely ordinary group object in the ordinary category of simplicial sets. At least the equivalence between pointed connected spaces and simplicial groups is the oldest of all these results-it goes back to Kan in the early '60s.
Anyway, hopefully that's given a bit more of an idea of what's going on. There are many approaches to the concept, largely because all of the approaches become intolerably complicated in one way or another. This situation is characteristic of $infty$-category theory, and in the current state of knowledge it appears unavoidable that appending "$infty$" to a familiar object creates substantial complications.
answered 1 hour ago
Kevin Carlson
31.4k23269
answered 1 hour ago
Kevin Carlson
31.4k23269
answered 1 hour ago
Kevin Carlson
31.4k23269
answered 1 hour ago
Kevin Carlson
31.4k23269
31.4k23269
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