Monoidal products, monoids and cosimplicial objects (Godement III)
In the last blog post I promised to introduce monads. Now that I have whetted your appetite, I will pull a bait and switch. We will not talk about monads; rather I will set the stage with some abstract nonsense. There is a famous definition of monad as a "monoid in the category of endofunctors $[\mathcal{C},\mathcal{C}]$ of some category $\mathcal{C}$." Before it is possible to parse this definition, we must understand what a monoid is. Along the way we will start to pick up the significance and substance of this definition, and why it should matter. There is a close connection between monoids and cosimplicial objects, which is what this post is really about; so before we even talk about monads properly, we will develop an inkling of why they would be relevant to simplicial theory.
Be warned before reading that this is very heady stuff. It is something I avoided learning for a long time, as the section in Categories for the Working Mathematician on this material I found to be impossibly intimidating and did not dare to look upon it. Nevertheless I have tried to make this readable.
One of my motivations for learning simplicial theory was to answer the question: Why do simplicial objects seem to arise everywhere? I have found an answer which is largely satisfying to me: Every monoid gives rise to a cosimplicial set. As monoids are of course some of the most ubiquitous objects in mathematics, it is no surprise that cosimplicial sets are just as ubiquitous.
Namely, if $M$ is a monoid (by which we mean a plain old monoid in the category of sets), then one can define an (augmented) cosimplicial object $X$ where $X_{-1} = \left\{ \ast \right\}$, $X_0 = M$, $X_1= M\times M$, and $X_n = M^{n+1}$, the $n+1$-fold Cartesian product; the degeneracy maps $s_i : M^n\to M^{n-1}$ send $(m_1,\dots, m_n)$ to $(m_1,\dots, m_{i-1},m_i\cdot m_{i+1},m_{i+2},\dots, m_n)$; the face maps $d_{i}: M^n\to M^{n+1}$ send $(m_1,\dots, m_n)$ to $(m_1,\dots, m_{i-1},e,m_i,\dots, m_n)$ where $e$ is the identity element of the monoid.
A similar construction works for algebras $A$ of a ring $R$; working in the category of $R$-modules, any algebra $A$ gives rise to an augmented cosimplicial $R$-module where $X_{-1}=R, X_0=A$, $X_1= A\otimes_RA$, and $X_n= A^{\otimes n+1}$ is the $n+1$-fold tensor product of $A$ with itself. The degeneracy maps $s_i: A^{\otimes n}\to A^{\otimes n-1}$ are given by $a_1\otimes a_2\dots\otimes a_n\mapsto a_1\otimes\dots\otimes a_ia_{i+1}\otimes\dots a_n$. The face maps $d_i: A^{\otimes n}\to A^{\otimes n+1}$ send $a_1\otimes\dots\otimes a_n\mapsto a_1\otimes\dots a_{i-1}\otimes 1\otimes a_{i}\dots\otimes a_n$.
A third example: Let $\mathcal{C}$ be the category of topological spaces. The join of two topological spaces $X$ and $Y$ is a cylinder-like construction which returns the space spanned by all straight lines joining a point in $X$ to a point in $Y$; see the first chapter of Hatcher's Algebraic Topology for an explanation. Formally we define $X\ast Y$ as $X\coprod (X\times I \times Y)\coprod Y/\sim$, where $I$ is the unit interval and where $\sim$ is the smallest equivalence relation which results from identifying $(x,0,y)$ with $x$ and $(x,1,y)$ with $y$. Observe that if $\left\{ \bullet \right\}$ represents the one-point space, then $\left\{ \bullet \right\}\ast\left\{ \bullet \right\} \cong I$, and more generally the $n$-fold join of the singleton space with itself gives the $(n-1)$-simplex. Interestingly enough, although a singleton does not much look like a monoid, the canonical maps $e:\emptyset\to \left\{ \bullet \right\}$ and $m : I\to \left\{ \bullet \right\}$ bear a sufficient formal similarity to the unit and multiplication map of a monoid that one can construct an (augmented) cosimplicial object in Top in an analogous way to the schema above. This simplicial object has $X_{[n]} = \Delta^{n}$, the $n$-simplex (here we will have to interpret $X_{-1}=\Delta^{-1}$ as the empty space; the face maps $d_i : \Delta^{n-1}\to \Delta^{n}$ are the inclusion maps of the $i$-th face and suspension maps $s_i: \Delta^{n+1}\to \Delta^n$ which are the maps which collaps the edge $i,i+1$ down to a point, crushing the faces $\left\{ v_0,\dots, v_{i-1},v_i,v_{i+2},\dots,v_{n+1}\right\}$ and $\left\{ v_0,\dots,v_{i-1},v_{i+1},v_{i+2},\dots, v_{n+1}\right\}$ down to a single $n$-dimensional face.
A similar construction can be designed in any category $\mathcal{C}$ carrying a notion of ''product'' $\otimes$ which is sufficiently similar in its formal properties to the Cartesian product of sets, the tensor product of modules, or the join of topological spaces - a monoidal product. We will not concern ourselves with the formal definition, which involves subtle coherence conditions, but in particular, the product $\otimes$ should be:
The name monoidal product is a bit of a double entendre. On the one hand, it is monoidal in the sense that (if the phrase "up to isomorphism'' is suppressed across the board) it appears to enrich the class of objects of $\mathcal{C}$ with the structure of a monoid, with multiplication given by $\otimes$. On the other hand, it is monoidal in that in order to try and give a definition of "monoid'' in an arbitrary category $\mathcal{C}$ which is general enough to incorporate both a monoid in $\mathbf{Sets}$ and an algebra over a ring in $R-mod$, as well as to carry out the above construction of the cosimplicial object associated to a monoid, one needs a category $\mathcal{C}$ equipped with a monoidal product $\otimes$.
Note that it is clear that the unit $1$ is determined up to isomorphism; as $1\cong 1\otimes 1'\cong 1'$.
Definition: A monoid in a monoidal category $(\mathcal{C},\otimes, 1)$ is an object $M\in Ob(\mathcal{C})$ equipped with maps $e : 1\to M$, called the unit of the monoid, and $m : M\otimes M\to M$, called the multiplication of the monoid, satisfying the unit law
\begin{equation}
\label{eq:184}
m\circ (e \otimes id_M)\circ \nu_M = m\circ (id_M\otimes e)\circ \rho_M = id_M
\end{equation} as maps $M\to M$ and the associativity law
\begin{equation}
\label{eq:185}
m\circ (m\otimes id_{M}) = m\circ (id_M\otimes m)\circ \alpha_{M,M,M}
\end{equation} as maps $(M\otimes M)\otimes M\to M$.
Note in order for us to even formulate this definition, we need to employ the natural isomorphisms $1\otimes M\cong M$ and $M\cong M\otimes 1$, as well as $M\otimes (M\otimes M)\cong (M\otimes M)\otimes M$. The associativity law and unit law for a monoid cannot even be formulated without the associativity and unitality of the monoidal product.
It can now be verified that if $M$ is a monoid in $(\mathcal{C},\otimes, 1)$, then there is an augmented cosimplicial object in $\mathcal{C}$ given by $X_{-1}=1,X_0=M,$ and $X_n = M\otimes (M\otimes (\dots \otimes M)\dots)$ whose degeneracy maps are given by $s_i : M^{\otimes n}\to M^{\otimes n-1}$ is exactly $id_M\otimes id_M\otimes\dots m\otimes\dots id_M$ with an $m : M\otimes M\to M$ in the $i$-th position, and with face maps $d_i: M^{\otimes n}\to M^{\otimes n+1}$ given by $id_M\otimes\dots \otimes e\otimes\dots M$, with a unit map $e:1\to M$ inserted in the $i$th position. (Here you must excuse me for dropping the associativity and unitality isomorphisms for the sake of readability; for example in the definition of $s_i$ there must be an identification of $M\otimes \dots\otimes M$ with $M\otimes \dots\otimes 1\otimes\dots\otimes M$.)
Although the reader may verify for themselves or gladly take it on faith that this does in fact define a cosimplicial object, they may still hold some skepticism that this is a natural construction to carry out. What is the meaning of this object? Why are we interested in it? The answer comes from following the category theorists' impulse to seek "free structures'' and universal properties. The reader is surely familiar with the free group, free vector space, and so on. It is in this light that we can recognize the simplex category, and the notion of cosimplicial object, as fundamental in the study of monoids.
The universal property of the free group $F(X)$ on a set $X$ is that whenever a group $G$ is given, and whenever we have a set map $f: X\to G$, then this map $f$ extends, uniquely, to a group homomorphism $f' : F(X)\to G$. The universal property of the free real vector space $\bigoplus_{x\in X}\mathbb{R}$ on a set $X$ is that for any set map $f: X\to V$ into a vector space $V$, there is a unique linear transformation $f': \bigoplus_{x\in X}\mathbb{R}\to V$ extending $f$ along the canonical embedding $X\to \bigoplus_{x\in X}\mathbb{R}$ sending $x$ to $1_x\in \mathbb{R}_x$. The cosimplicial object associated to a monoid in a monoidal category is the free categorical structure generated by the free application of the monoidal product functor, the multiplication map, and the unit map, in a sense we will now make precise.
In order to explain the universal property of the simplex category, let us make some simplifying assumptions. If $(\mathcal{C},\otimes,1,\alpha,\nu,\rho)$ is a monoidal category, we will say that the monoidal product is strict if
I will abbreviate "strict monoidal category'' as SMC.
Because we are most familiar with universal algebras generated by a set, in order to better illustrate our point, we will prefer to regard an SMC as a two-sorted algebraic structure. Let us, therefore, write down a definition of an SMC in a way that is more reminiscent of group theory or some other traditionally algebraic field.
An SMC is a two-sorted algebraic structure, a pair of sets $(Ob(\mathcal{C}),Arr(\mathcal{C}))$, together with the following additional structure:
satisfying the following axioms:
An SMC with distinguished choice of monoid is precisely an SMC in the above sense equipped with the following additional structure:
In this vein, the monoidal product $\otimes$ can be identified with the ordinal addition $+$ which identifies $[n]+[m]$ with $[n+m+1]$; the unit $1$ is the empty ordinal $\emptyset=[-1]$; the monoid $(M, e: 1\to M, m: M\otimes M\to M)$ can be identified with $([0],e : [-1]\to [0],m: [1]\to [0])$, where both $e$ and $m$ are the unique maps ($[0]$ is the terminal object in this category.)
One may prove this fact by appeal to an unpleasant induction on term complexity. A more elegant argument can be carried out in the language of string diagrams, which is a beautiful visual language and calculus for working in a monoidal category whose simplicity and ease of use for such problems is far beyond that of first-order logic. This language renders standard algebraic reasoning obsolete in many cases where it applies.
We introduce now a category SMC of strict monoidal categories. An object $(\mathcal{C},\otimes, 1)$ in SMC is a category $\mathcal{C}$ equipped with a strict monoidal product $\otimes$ and a unit $1$. A morphism of SMC's is a functor $F: (\mathcal{C},\otimes_{\mathcal{C}},1_{\mathcal{C}})\to (\mathcal{D},\otimes_{\mathcal{D}},1_{\mathcal{D}})$ such that $F(1_{\mathcal{C}})=1_{\mathcal{D}}$ and $F(A\otimes_{\mathcal{C}}B) = F(A)\otimes_{\mathcal{D}}F(B)$ on the nose. No doubt such a functor corresponds to a "homomorphism'' of SMC's under whatever general formulation of homomorphism of algebras of a multi-sorted first-order algebraic theory the reader wants to reference, in their favorite text on universal algebra or model theory.
One can then define a category of SMC's equipped with a distinguished choice of monoid; and a morphism of such categories $(\mathcal{C},\otimes_{\mathcal{C}},1_{\mathcal{C}},M_{\mathcal{C}},e_{\mathcal{C}},m_{\mathcal{C}})$ to $(\mathcal{D},\dots)$ would be a morphism of SMC's which also carries $M_{\mathcal{C}}$ to $M_{\mathcal{D}},$ carries $e_{\mathcal{C}}$ to $e_{\mathcal{D}}$, and carries $m_{\mathcal{C}}$ to $m_{\mathcal{D}}$.
Our logical-syntactic claim that the augmented simplex category is the "free SMC equipped with distinguished monoid'' can be translated into categorical language by saying that the simplex category equipped with the distinguished monoid $([0],e: [-1]\to [0],m: [1]\to [0])$ is the initial object in the category of SMC's with distinguished monoid and functors between SMC's preserving the monoidal product on the nose and carrying distinguished monoids to distinguished monoid. Put another way: In any given SMC $(C,\otimes, 1)$, for any monoid $(M,e: 1\to M, m: M\otimes M\to M)$ there is a unique augmented cosimplicial object $X: \Delta_a\to \mathcal{C}$ preserving the unit $1$, satisfying $X_{[n]}\otimes X_{[m]} = X_{[n+m+1]}$, and carrying $[0]$ to $M$, $e: [-1]\to [0]$ to $e' : 1\to M$, and $m : [1]\to [0]$ to $m': M\otimes M\to M$. Conversely, every cosimplicial object in $\mathcal{C}$ preserving the unit and monoidal product on-the-nose defines such a monoid.
It is this theorem, which, in my opinion, explains the prevalence of simplicial objects throughout mathematics.
Be warned before reading that this is very heady stuff. It is something I avoided learning for a long time, as the section in Categories for the Working Mathematician on this material I found to be impossibly intimidating and did not dare to look upon it. Nevertheless I have tried to make this readable.
One of my motivations for learning simplicial theory was to answer the question: Why do simplicial objects seem to arise everywhere? I have found an answer which is largely satisfying to me: Every monoid gives rise to a cosimplicial set. As monoids are of course some of the most ubiquitous objects in mathematics, it is no surprise that cosimplicial sets are just as ubiquitous.
Namely, if $M$ is a monoid (by which we mean a plain old monoid in the category of sets), then one can define an (augmented) cosimplicial object $X$ where $X_{-1} = \left\{ \ast \right\}$, $X_0 = M$, $X_1= M\times M$, and $X_n = M^{n+1}$, the $n+1$-fold Cartesian product; the degeneracy maps $s_i : M^n\to M^{n-1}$ send $(m_1,\dots, m_n)$ to $(m_1,\dots, m_{i-1},m_i\cdot m_{i+1},m_{i+2},\dots, m_n)$; the face maps $d_{i}: M^n\to M^{n+1}$ send $(m_1,\dots, m_n)$ to $(m_1,\dots, m_{i-1},e,m_i,\dots, m_n)$ where $e$ is the identity element of the monoid.
A similar construction works for algebras $A$ of a ring $R$; working in the category of $R$-modules, any algebra $A$ gives rise to an augmented cosimplicial $R$-module where $X_{-1}=R, X_0=A$, $X_1= A\otimes_RA$, and $X_n= A^{\otimes n+1}$ is the $n+1$-fold tensor product of $A$ with itself. The degeneracy maps $s_i: A^{\otimes n}\to A^{\otimes n-1}$ are given by $a_1\otimes a_2\dots\otimes a_n\mapsto a_1\otimes\dots\otimes a_ia_{i+1}\otimes\dots a_n$. The face maps $d_i: A^{\otimes n}\to A^{\otimes n+1}$ send $a_1\otimes\dots\otimes a_n\mapsto a_1\otimes\dots a_{i-1}\otimes 1\otimes a_{i}\dots\otimes a_n$.
A third example: Let $\mathcal{C}$ be the category of topological spaces. The join of two topological spaces $X$ and $Y$ is a cylinder-like construction which returns the space spanned by all straight lines joining a point in $X$ to a point in $Y$; see the first chapter of Hatcher's Algebraic Topology for an explanation. Formally we define $X\ast Y$ as $X\coprod (X\times I \times Y)\coprod Y/\sim$, where $I$ is the unit interval and where $\sim$ is the smallest equivalence relation which results from identifying $(x,0,y)$ with $x$ and $(x,1,y)$ with $y$. Observe that if $\left\{ \bullet \right\}$ represents the one-point space, then $\left\{ \bullet \right\}\ast\left\{ \bullet \right\} \cong I$, and more generally the $n$-fold join of the singleton space with itself gives the $(n-1)$-simplex. Interestingly enough, although a singleton does not much look like a monoid, the canonical maps $e:\emptyset\to \left\{ \bullet \right\}$ and $m : I\to \left\{ \bullet \right\}$ bear a sufficient formal similarity to the unit and multiplication map of a monoid that one can construct an (augmented) cosimplicial object in Top in an analogous way to the schema above. This simplicial object has $X_{[n]} = \Delta^{n}$, the $n$-simplex (here we will have to interpret $X_{-1}=\Delta^{-1}$ as the empty space; the face maps $d_i : \Delta^{n-1}\to \Delta^{n}$ are the inclusion maps of the $i$-th face and suspension maps $s_i: \Delta^{n+1}\to \Delta^n$ which are the maps which collaps the edge $i,i+1$ down to a point, crushing the faces $\left\{ v_0,\dots, v_{i-1},v_i,v_{i+2},\dots,v_{n+1}\right\}$ and $\left\{ v_0,\dots,v_{i-1},v_{i+1},v_{i+2},\dots, v_{n+1}\right\}$ down to a single $n$-dimensional face.
A similar construction can be designed in any category $\mathcal{C}$ carrying a notion of ''product'' $\otimes$ which is sufficiently similar in its formal properties to the Cartesian product of sets, the tensor product of modules, or the join of topological spaces - a monoidal product. We will not concern ourselves with the formal definition, which involves subtle coherence conditions, but in particular, the product $\otimes$ should be:
- functorial, in that it determines a functor $\mathcal{C}\times \mathcal{C}\to \mathcal{C}$ sending $(A,B)$ to $A\otimes B$
- associative up to a natural isomorphism $\alpha$, so that the two functors $(A,B,C)\mapsto (A\otimes B)\otimes C$ and $(A,B,C)\mapsto A\otimes (B\otimes C)$ are naturally isomorphic in $[\mathcal{C}^3,\mathcal{C}]$ by a certain distinguished choice of natural isomorphism $\alpha$
- unital up to isomorphism, so that there is a distinguished object $1\in \mathcal{C}$, which we call the unit of the monoidal product, such that the functors $1\otimes - :\mathcal{C}\to \mathcal{C}$ and $-\otimes 1: \mathcal{C}\to \mathcal{C}$ are both naturally isomorphic to the identity functor $id: \mathcal{C}\to \mathcal{C}$, by two distinguished choices $\rho : id_{\mathcal{C}}\to -\otimes 1$ and $\nu : id_{\mathcal{C}}\to 1\otimes -$.
The name monoidal product is a bit of a double entendre. On the one hand, it is monoidal in the sense that (if the phrase "up to isomorphism'' is suppressed across the board) it appears to enrich the class of objects of $\mathcal{C}$ with the structure of a monoid, with multiplication given by $\otimes$. On the other hand, it is monoidal in that in order to try and give a definition of "monoid'' in an arbitrary category $\mathcal{C}$ which is general enough to incorporate both a monoid in $\mathbf{Sets}$ and an algebra over a ring in $R-mod$, as well as to carry out the above construction of the cosimplicial object associated to a monoid, one needs a category $\mathcal{C}$ equipped with a monoidal product $\otimes$.
Note that it is clear that the unit $1$ is determined up to isomorphism; as $1\cong 1\otimes 1'\cong 1'$.
Definition: A monoid in a monoidal category $(\mathcal{C},\otimes, 1)$ is an object $M\in Ob(\mathcal{C})$ equipped with maps $e : 1\to M$, called the unit of the monoid, and $m : M\otimes M\to M$, called the multiplication of the monoid, satisfying the unit law
\begin{equation}
\label{eq:184}
m\circ (e \otimes id_M)\circ \nu_M = m\circ (id_M\otimes e)\circ \rho_M = id_M
\end{equation} as maps $M\to M$ and the associativity law
\begin{equation}
\label{eq:185}
m\circ (m\otimes id_{M}) = m\circ (id_M\otimes m)\circ \alpha_{M,M,M}
\end{equation} as maps $(M\otimes M)\otimes M\to M$.
Note in order for us to even formulate this definition, we need to employ the natural isomorphisms $1\otimes M\cong M$ and $M\cong M\otimes 1$, as well as $M\otimes (M\otimes M)\cong (M\otimes M)\otimes M$. The associativity law and unit law for a monoid cannot even be formulated without the associativity and unitality of the monoidal product.
It can now be verified that if $M$ is a monoid in $(\mathcal{C},\otimes, 1)$, then there is an augmented cosimplicial object in $\mathcal{C}$ given by $X_{-1}=1,X_0=M,$ and $X_n = M\otimes (M\otimes (\dots \otimes M)\dots)$ whose degeneracy maps are given by $s_i : M^{\otimes n}\to M^{\otimes n-1}$ is exactly $id_M\otimes id_M\otimes\dots m\otimes\dots id_M$ with an $m : M\otimes M\to M$ in the $i$-th position, and with face maps $d_i: M^{\otimes n}\to M^{\otimes n+1}$ given by $id_M\otimes\dots \otimes e\otimes\dots M$, with a unit map $e:1\to M$ inserted in the $i$th position. (Here you must excuse me for dropping the associativity and unitality isomorphisms for the sake of readability; for example in the definition of $s_i$ there must be an identification of $M\otimes \dots\otimes M$ with $M\otimes \dots\otimes 1\otimes\dots\otimes M$.)
Although the reader may verify for themselves or gladly take it on faith that this does in fact define a cosimplicial object, they may still hold some skepticism that this is a natural construction to carry out. What is the meaning of this object? Why are we interested in it? The answer comes from following the category theorists' impulse to seek "free structures'' and universal properties. The reader is surely familiar with the free group, free vector space, and so on. It is in this light that we can recognize the simplex category, and the notion of cosimplicial object, as fundamental in the study of monoids.
The universal property of the free group $F(X)$ on a set $X$ is that whenever a group $G$ is given, and whenever we have a set map $f: X\to G$, then this map $f$ extends, uniquely, to a group homomorphism $f' : F(X)\to G$. The universal property of the free real vector space $\bigoplus_{x\in X}\mathbb{R}$ on a set $X$ is that for any set map $f: X\to V$ into a vector space $V$, there is a unique linear transformation $f': \bigoplus_{x\in X}\mathbb{R}\to V$ extending $f$ along the canonical embedding $X\to \bigoplus_{x\in X}\mathbb{R}$ sending $x$ to $1_x\in \mathbb{R}_x$. The cosimplicial object associated to a monoid in a monoidal category is the free categorical structure generated by the free application of the monoidal product functor, the multiplication map, and the unit map, in a sense we will now make precise.
In order to explain the universal property of the simplex category, let us make some simplifying assumptions. If $(\mathcal{C},\otimes,1,\alpha,\nu,\rho)$ is a monoidal category, we will say that the monoidal product is strict if
- $A\otimes (B\otimes C) = (A\otimes B)\otimes C$ on the nose, and $\alpha$ is the identity natural transformation on that functor; and furthermore
- $1\otimes A = A\otimes 1 = A$ on the nose, and the natural transformations $\nu$ and $\rho$ are the identity natural transformations.
I will abbreviate "strict monoidal category'' as SMC.
Because we are most familiar with universal algebras generated by a set, in order to better illustrate our point, we will prefer to regard an SMC as a two-sorted algebraic structure. Let us, therefore, write down a definition of an SMC in a way that is more reminiscent of group theory or some other traditionally algebraic field.
An SMC is a two-sorted algebraic structure, a pair of sets $(Ob(\mathcal{C}),Arr(\mathcal{C}))$, together with the following additional structure:
- two maps $dom,cod : Arr(\mathcal{C})\to Ob(\mathcal{C})$, a map $id: Ob(\mathcal{C})\to Arr(\mathcal{C})$
- a partially defined map $\circ : Arr(\mathcal{C})\times Arr(\mathcal{C})\to Arr(\mathcal{C})$, where $g\circ f$ is defined iff $dom(g) = cod(f)$
- a pair of maps $\otimes : Ob(\mathcal{C})\times Ob(\mathcal{C})\to Ob(\mathcal{C})$, $\otimes : Arr(\mathcal{C})\times Arr(\mathcal{C})\to Arr(\mathcal{C})$
- a distinguished element $1\in Ob(\mathcal{C})$
satisfying the following axioms:
- $dom(id(B)) = cod(id(B)) = B$ for all $B\in Ob(\mathcal{C})$
- $f\circ id(dom(f))) = id(cod(f)\circ f = f$ for all $f\in Arr(\mathcal{C})$
- $f\circ (g\circ h) = (f\circ g)\circ h$ whenever this is defined
- $(f\circ g) \otimes (f' \circ g') = (f\otimes f')\circ (g\otimes g')$ whenever this is defined
- $dom(f \otimes g) = dom(f)\otimes dom(g)$, $cod(f \otimes g) = cod(f)\otimes cod(g)$
- $id(A)\otimes id(B)= id(A\otimes B)$
- $A\otimes (B \otimes C) = (A\otimes B)\otimes C$, all $A,B,C\in Ob(\mathcal{C})$
- $(f\otimes g)\otimes h = f\otimes (g\otimes h)$ for all $f,g,h\in Arr(\mathcal{C})$
- $1\otimes B = B\otimes 1= B$ for all $B\in Ob(\mathcal{C})$
An SMC with distinguished choice of monoid is precisely an SMC in the above sense equipped with the following additional structure:
- a distinguished element $M\in Ob(\mathcal{C})$
- a pair $m,e \in Arr(\mathcal{C})$
- $dom(m) = M\otimes M; cod(m) = M$
- $dom(e) = 1, cod(e) =M$
- $m\circ (e\otimes id(M))= id(M) = m\circ (id(M)\otimes e)$
- $m\circ (id(M)\otimes m)= m\circ (m\otimes id(M))$
In this vein, the monoidal product $\otimes$ can be identified with the ordinal addition $+$ which identifies $[n]+[m]$ with $[n+m+1]$; the unit $1$ is the empty ordinal $\emptyset=[-1]$; the monoid $(M, e: 1\to M, m: M\otimes M\to M)$ can be identified with $([0],e : [-1]\to [0],m: [1]\to [0])$, where both $e$ and $m$ are the unique maps ($[0]$ is the terminal object in this category.)
One may prove this fact by appeal to an unpleasant induction on term complexity. A more elegant argument can be carried out in the language of string diagrams, which is a beautiful visual language and calculus for working in a monoidal category whose simplicity and ease of use for such problems is far beyond that of first-order logic. This language renders standard algebraic reasoning obsolete in many cases where it applies.
We introduce now a category SMC of strict monoidal categories. An object $(\mathcal{C},\otimes, 1)$ in SMC is a category $\mathcal{C}$ equipped with a strict monoidal product $\otimes$ and a unit $1$. A morphism of SMC's is a functor $F: (\mathcal{C},\otimes_{\mathcal{C}},1_{\mathcal{C}})\to (\mathcal{D},\otimes_{\mathcal{D}},1_{\mathcal{D}})$ such that $F(1_{\mathcal{C}})=1_{\mathcal{D}}$ and $F(A\otimes_{\mathcal{C}}B) = F(A)\otimes_{\mathcal{D}}F(B)$ on the nose. No doubt such a functor corresponds to a "homomorphism'' of SMC's under whatever general formulation of homomorphism of algebras of a multi-sorted first-order algebraic theory the reader wants to reference, in their favorite text on universal algebra or model theory.
One can then define a category of SMC's equipped with a distinguished choice of monoid; and a morphism of such categories $(\mathcal{C},\otimes_{\mathcal{C}},1_{\mathcal{C}},M_{\mathcal{C}},e_{\mathcal{C}},m_{\mathcal{C}})$ to $(\mathcal{D},\dots)$ would be a morphism of SMC's which also carries $M_{\mathcal{C}}$ to $M_{\mathcal{D}},$ carries $e_{\mathcal{C}}$ to $e_{\mathcal{D}}$, and carries $m_{\mathcal{C}}$ to $m_{\mathcal{D}}$.
Our logical-syntactic claim that the augmented simplex category is the "free SMC equipped with distinguished monoid'' can be translated into categorical language by saying that the simplex category equipped with the distinguished monoid $([0],e: [-1]\to [0],m: [1]\to [0])$ is the initial object in the category of SMC's with distinguished monoid and functors between SMC's preserving the monoidal product on the nose and carrying distinguished monoids to distinguished monoid. Put another way: In any given SMC $(C,\otimes, 1)$, for any monoid $(M,e: 1\to M, m: M\otimes M\to M)$ there is a unique augmented cosimplicial object $X: \Delta_a\to \mathcal{C}$ preserving the unit $1$, satisfying $X_{[n]}\otimes X_{[m]} = X_{[n+m+1]}$, and carrying $[0]$ to $M$, $e: [-1]\to [0]$ to $e' : 1\to M$, and $m : [1]\to [0]$ to $m': M\otimes M\to M$. Conversely, every cosimplicial object in $\mathcal{C}$ preserving the unit and monoidal product on-the-nose defines such a monoid.
It is this theorem, which, in my opinion, explains the prevalence of simplicial objects throughout mathematics.
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