nLab geometry of physics - basic notions of categorical algebra

Basic notions of Categorical algebra

We have seen that the existence of Cartesian products in a category $\mathcal{C}$ equips is with a functor of the form

$$\mathcal{C} \times \mathcal{C} \overset{ (-) \times (-) }{\longrightarrow} \mathcal{C}$$

which is directly analogous to the operation of multiplication in an associative algebra or even just in a semigroup (or monoid), just “categorified” (Example below). This is made precise by the concept of a monoidal category (Def. below).

This relation between category theory and algebra leads to the fields of categorical algebra and of universal algebra.

Here we are mainly interested in monoidal categories as a foundations for enriched category theory, to which we turn below.

$\,$

Monoidal categories

Definition

(monoidal category)

An_monoidal category_ is a category $\mathcal{C}$ (Def. ) equipped with

1. a functor (Def. )

   $$\otimes \;\colon\; \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C}$$

   out of the product category of $\mathcal{C}$ with itself (Example ), called the tensor product,

2. an object

   $$1 \in Obj_{\mathcal{C}}$$

   called the unit object or tensor unit,

3. a natural isomorphism (Def. )

   $$a \;\colon\; ((-)\otimes (-)) \otimes (-) \overset{\simeq}{\longrightarrow} (-) \otimes ((-)\otimes(-))$$

   called the associator,

4. a natural isomorphism

   $$\ell \;\colon\; (1 \otimes (-)) \overset{\simeq}{\longrightarrow} (-)$$

   called the left unitor, and a natural isomorphism

   $$r \;\colon\; (-) \otimes 1 \overset{\simeq}{\longrightarrow} (-)$$

   called the right unitor,

such that the following two kinds of diagrams commute, for all objects involved:

1. triangle identity:

   $$\array{ & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y) \\ & {}_{\rho_x \otimes 1_y}\searrow && \swarrow_{1_x \otimes \lambda_y} & \\ && x \otimes y && }$$

2. the pentagon identity:

   $$\array{ && (w \otimes x) \otimes (y \otimes z) \\ & {}^{\mathllap{\alpha_{w \otimes x, y, z}}}\nearrow && \searrow^{\mathrlap{\alpha_{w,x,y \otimes z}}} \\ ((w \otimes x ) \otimes y) \otimes z && && (w \otimes (x \otimes (y \otimes z))) \\ {}^{\mathllap{\alpha_{w,x,y}} \otimes id_z }\downarrow && && \uparrow^{\mathrlap{ id_w \otimes \alpha_{x,y,z} }} \\ (w \otimes (x \otimes y)) \otimes z && \underset{\alpha_{w,x \otimes y, z}}{\longrightarrow} && w \otimes ( (x \otimes y) \otimes z ) }$$

Example

(cartesian monoidal category)

Let $\mathcal{C}$ be a category in which all finite products exist. Then $\mathcal{C}$ becomes a monoidal category (Def. ) by

1. taking the tensor product to be the Cartesian product

   $$X \otimes Y \;\coloneqq\; X \times Y$$

2. taking the unit object to be the terminal object (Def. )

   $$I \;\coloneqq\; \ast$$

Monoidal categories of this form are called cartesian monoidal categories.

Lemma

(Kelly 64)

Let $(\mathcal{C}, \otimes, 1)$ be a monoidal category, def. . Then the left and right unitors $\ell$ and $r$ satisfy the following conditions:

1. $\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1$;

2. for all objects $x,y \in \mathcal{C}$ the following diagrams commutes:

   $$\array{ (1 \otimes x) \otimes y & & \\ {}^\mathllap{\alpha_{1, x, y}} \downarrow & \searrow^\mathrlap{\ell_x \otimes id_y} & \\ 1 \otimes (x \otimes y) & \underset{\ell_{x \otimes y}}{\longrightarrow} & x \otimes y } \,;$$

   and

   $$\array{ x \otimes (y \otimes 1) & & \\ {}^\mathllap{\alpha^{-1}_{1, x, y}} \downarrow & \searrow^\mathrlap{id_x \otimes r_y} & \\ (x \otimes y) \otimes 1 & \underset{r_{x \otimes y}}{\longrightarrow} & x \otimes y } \,;$$

For proof see at monoidal category this lemma and this lemma.

Remark

Just as for an associative algebra it is sufficient to demand $1 a = a$ and $a 1 = a$ and $(a b) c = a (b c)$ in order to have that expressions of arbitrary length may be re-bracketed at will, so there is a coherence theorem for monoidal categories which states that all ways of freely composing the unitors and associators in a monoidal category (def. ) to go from one expression to another will coincide. Accordingly, much as one may drop the notation for the bracketing in an associative algebra altogether, so one may, with due care, reason about monoidal categories without always making all unitors and associators explicit.

(Here the qualifier “freely” means informally that we must not use any non-formal identification between objects, and formally it means that the diagram in question must be in the image of a strong monoidal functor from a free monoidal category. For example if in a particular monoidal category it so happens that the object $X \otimes (Y \otimes Z)$ is actually equal to $(X \otimes Y)\otimes Z$, then the various ways of going from one expression to another using only associators and this equality no longer need to coincide.)

Definition

(braided monoidal category)

A braided monoidal category, is a monoidal category $\mathcal{C}$ (def. ) equipped with a natural isomorphism (Def. )

(1)$$\tau_{x,y} \colon x \otimes y \to y \otimes x$$

called the braiding, such that the following two kinds of diagrams commute for all objects involved (“hexagon identities”):

$$\array{ (x \otimes y) \otimes z &\stackrel{a_{x,y,z}}{\to}& x \otimes (y \otimes z) &\stackrel{\tau_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{\tau_{x,y}\otimes Id} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{Id \otimes \tau_{x,z}}{\to}& y \otimes (z \otimes x) }$$

and

$$\array{ x \otimes (y \otimes z) &\stackrel{a^{-1}_{x,y,z}}{\to}& (x \otimes y) \otimes z &\stackrel{\tau_{x \otimes y, z}}{\to}& z \otimes (x \otimes y) \\ \downarrow^{Id \otimes \tau_{y,z}} &&&& \downarrow^{a^{-1}_{z,x,y}} \\ x \otimes (z \otimes y) &\stackrel{a^{-1}_{x,z,y}}{\to}& (x \otimes z) \otimes y &\stackrel{\tau_{x,z} \otimes Id}{\to}& (z \otimes x) \otimes y } \,,$$

where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the associator of $\mathcal{C}^\otimes$.

Definition

A symmetric monoidal category is a braided monoidal category (def. ) for which the braiding

$$\tau_{x,y} \colon x \otimes y \to y \otimes x$$

satisfies the condition:

$$\tau_{y,x} \circ \tau_{x,y} = 1_{x \otimes y}$$

for all objects $x, y$

Remark

In analogy to the coherence theorem for monoidal categories (remark ) there is a coherence theorem for symmetric monoidal categories (def. ), saying that every diagram built freely (see remark ) from associators, unitors and braidings such that both sides of the diagram correspond to the same permutation of objects, coincide.

Definition

(symmetric closed monoidal category)

Given a symmetric monoidal category $\mathcal{C}$ with tensor product $\otimes$ (def. ) it is called a closed monoidal category if for each $Y \in \mathcal{C}$ the functor $Y \otimes(-)\simeq (-)\otimes Y$ has a right adjoint, denoted $hom(Y,-)$

(2)$$\mathcal{C} \underoverset {\underset{ [Y,-]}{\longrightarrow}} {\overset{(-) \otimes Y}{\longleftarrow}} {\bot} \mathcal{C} \,,$$

hence if there are natural bijections

$$Hom_{\mathcal{C}}(X \otimes Y, Z) \;\simeq\; Hom_{\mathcal{C}}{C}(X, [Y,Z])$$

for all objects $X,Z \in \mathcal{C}$.

Since for the case that $X = 1$ is the tensor unit of $\mathcal{C}$ this means that

$$Hom_{\mathcal{C}}(1, [Y,Z]) \simeq Hom_{\mathcal{C}}(Y,Z) \,,$$

the object $[Y,Z] \in \mathcal{C}$ is an enhancement of the ordinary hom-set $Hom_{\mathcal{C}}(Y,Z)$ to an object in $\mathcal{C}$. Accordingly, it is also called the internal hom between $Y$ and $Z$.

The adjunction counit (Def. ) in this case is called the evaluation morphism

(3)$$X \otimes [X,Y] \overset{ev}{\longrightarrow} Y$$

Example

(Set is a cartesian closed category)

The category Set of all sets (Example ) equipped with its cartesian monoidal category-structure (Example ) is a closed monoidal category (Def. ), hence a cartesian closed category. The Cartesian product is the original Cartesian product of sets, and the internal hom is the function set $[X,Y]$ of functions from $X$ to $Y$

Example

(tensor product of abelian groups is closed monoidal category symmetric monoidal category-structure)

The category Ab of abelian groups (as in Example ) becomes a symmetric monoidal category (Def. ) with tensor product the actual tensor product of abelian groups $\otimes_{\mathbb{Z}}$ and with tensor unit the additive group $\mathbb{Z}$ of integers. Again the associator, unitor and braiding isomorphism are the evident ones coming from the underlying sets.

This is a closed monoidal category with internal hom $hom(A,B)$ being the set of homomorphisms $Hom_{Ab}(A,B)$ equipped with the pointwise group structure for $\phi_1, \phi_2 \in Hom_{Ab}(A,B)$ then $(\phi_1 + \phi_2)(a) \coloneqq \phi_1(a) + \phi_2(b) \; \in B$.

This is the archetypical case that motivates the notation “$\otimes$” for the pairing operation in a monoidal category.

Example

(Cat and Grpd are cartesian closed categories)

The category Cat (Example ) of all small categories (Example ) is a cartesian monoidal category-structure (Example ) with Cartesian product given by forming product categories (Example ).

Inside this, the full subcategory (Example ) Grpd (Example ) of all small groupoids (Example ) is itself a cartesian monoidal category-structure (Example ) with Cartesian product given by forming product categories (Example ).

In both cases this yields a closed monoidal category (Def. ), hence a cartesian closed category: the internal hom is given by the functor category construction (Example ).

Example

(categories of presheaves are cartesian closed)

Let $\mathcal{C}$ be a category and write $[\mathcal{C}^{op}, Set]$ for its category of presheaves (Example ).

This is

1. a cartesian monoidal category (Example ), whose Cartesian product is given objectwise in $\mathcal{C}$ by the Cartesian product in Set:

   for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$, their Cartesian product $\mathbf{X} \times \mathbf{Y}$ exists and is given by

   $$\mathbf{X} \times \mathbf{Y} \;\;\colon\;\;\phantom{A} \array{ c_1 &\mapsto& \mathbf{X}(c_1) \times \mathbf{Y}(c_1) \\ {}^{\mathllap{f}}\big\downarrow && \big\uparrow^{ \mathrlap{ \mathbf{X}(f) \times \mathbf{Y}(f) } } \\ c_2 &\mapsto& \mathbf{X}(c_2) \times \mathbf{Y}(c_2) }$$

2. a cartesian closed category (Def. ), whose internal hom is given for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ by

   $$[\mathbf{X}, \mathbf{Y}] \;\;\colon\;\;\phantom{A} \array{ c_1 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_1) \times \mathbf{X}, \mathbf{y} ) \\ {}^{ \mathllap{ f } }\big\downarrow && \big\uparrow^{ \mathrlap{ Hom_{[\mathcal{C}^{op}, Set]}( y(f) \times \mathbf{X}, \mathbf{y} ) } } \\ c_2 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_2) \times \mathbf{X}, \mathbf{y} ) }$$

   Here $y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set]$ denotes the Yoneda embedding and $Hom_{[\mathcal{C}^{op}, Set]}(-,-)$ is the hom-functor on the category of presheaves.

Proof

The first statement is a special case of the general fact that limits of presheaves are computed objectwise (Example ).

For the second statement, first assume that $[\mathbf{X}, \mathbf{Y}]$ does exist. Then by the adjunction hom-isomorphism (?) we have for any other presheaf $\mathbf{Z}$ a natural isomorphism of the form

(4)$$Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z}, [\mathbf{X},\mathbf{Y}]) \;\simeq\; Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z} \times \mathbf{X}, \mathbf{Y}) \,.$$

This holds in particular for $\mathbf{Z} = y(c)$ a representable presheaf (Example ) and so the Yoneda lemma (Prop. ) implies that if it exists, then $[\mathbf{X}, \mathbf{Y}]$ must have the claimed form:

$$\begin{aligned} [\mathbf{X}, \mathbf{Y}](c) & \simeq Hom_{[\mathcal{C}^{op}, Set]}( y(c), [\mathbf{X}, \mathbf{Y}] ) \\ & \simeq Hom_{ [\mathcal{C}^{op}, Set] }( y(c) \times \mathbf{X}, \mathbf{Y} ) \,. \end{aligned}$$

Hence it remains to show that this formula does make (4) hold generally.

For this we use the equivalent characterization of adjoint functors from Prop. , in terms of the adjunction counit providing a system of universal arrows (Def. ).

Define a would-be adjunction counit, hence a would-be evaluation morphism (3), by

$$\array{ \mathbf{X} \times [\mathbf{X} , \mathbf{Y}] &\overset{ev}{\longrightarrow}& \mathbf{Y} \\ \mathbf{X}(c) \times Hom_{[\mathcal{C}^{op}, Set]}(y(c) \times \mathbf{X}, \mathbf{Y}) &\overset{ev_c}{\longrightarrow}& \mathbf{Y}(c) \\ (x, \phi) &\mapsto& \phi_c( id_c, x ) }$$

Then it remains to show that for every morphism of presheaves of the form $\mathbf{X} \times \mathbf{A} \overset{\phantom{A}f\phantom{A}}{\longrightarrow} \mathbf{Y}$ there is a unique morphism $\widetilde f \;\colon\; \mathbf{A} \longrightarrow [\mathbf{X}, \mathbf{Y}]$ such that

(5)$$\array{ \mathbf{X} \times \mathbf{A} && \overset{ \mathbf{X} \times \widetilde f }{\longrightarrow} && \mathbf{X} \times [\mathbf{X}, \mathbf{Y}] \\ & {}_{\mathllap{ \mathrlap{f} }}\searrow && \swarrow_{ \mathrlap{ ev } } \\ && \mathbf{Y} }$$

The commutativity of this diagram means in components at $c \in \mathcal{C}$ that, that for all $x \in \mathbf{X}(c)$ and $a \in \mathbf{A}(c)$ we have

$$\begin{aligned} ev_c( x, \widetilde f_c(a) ) & \coloneqq (\widetilde f_c(a))_c( id_c, x ) \\ & = f_c( x, a ) \end{aligned}$$

Hence this fixes the component $\widetilde f_c(a)_c$ when its first argument is the identity morphism $id_c$. But let $g \;\colon\; d \to c$ be any morphism and chase $(id_c, x )$ through the naturality diagram for $\widetilde f_c(a)$:

$$\array{ Hom_{\mathcal{C}}(c,c) \times \mathbf{X}(c) &\overset{ (\widetilde f_c(a))_c }{\longrightarrow}& \mathbf{Y}(c) \\ {}^{\mathllap{ g^\ast }}\big\downarrow && \big\downarrow^{\mathrlap{ g^\ast }} \\ Hom_{\mathcal{C}}(d,c) \times \mathbf{X}(d) &\overset{ (\widetilde f_c(a))_d }{\longrightarrow}& \mathbf{Y}(d) } \phantom{AAAA} \array{ \{ (id_c, x ) \} &\longrightarrow& \{ f_c( x, a ) \} \\ \big\downarrow && \big\downarrow \\ \{ (g, g^\ast(x)) \} &\longrightarrow& \{ f_d( g^\ast(x), g^\ast(a) ) \} }$$

This shows that $(\widetilde f_c(a))_d$ is fixed to be given by

(6)$$(\widetilde f_c(a))_d( g, x' ) \;=\; f_d( x', g^\ast(a) )$$

at least on those pairs $(g,x')$ such that $x'$ is in the image of $g^\ast$.

But, finally, $(\widetilde f_c(a))_d$ is also natural in $c$

$$\array{ \mathbf{A}(c) &\overset{ \widetilde f_c }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](c) \\ {}^{\mathllap{g^\ast}}\big\downarrow && \big\downarrow^{\mathrlap{g^\ast}} \\ \mathbf{A}(d) &\overset{ \widetilde f_d }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](d) }$$

which implies that (6) must hold generally. Hence naturality implies that (5) indeed has a unique solution.

$\,$

The internal hom (Def. ) turns out to share all the abstract properties of the ordinary (external) hom-functor (Def. ), even though this is not completely manifest from its definition. We make this explicit by the following three propositions.

Proposition

(internal hom bifunctor)

For $\mathcal{C}$ a closed monoidal category (Def. ), there is a unique functor (Def. ) out of the product category (Def. ) of $\mathcal{C}$ with its opposite category (Def. )

$$[-,-] \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{C}$$

such that for each $X \in \mathcal{C}$ it coincides with the internal hom $[X,-]$ (2) as a functor in the second variable, and such that there is a natural isomorphism

$$Hom(X, [Y,Z]) \;\simeq\; Hom(X \otimes Y, Z)$$

which is natural not only in $X$ and $Z$, but also in $Y$.

Proof

We have a natural isomorphism for each fixed $Y$, and hence in particular for fixed $Y$ and fixed $Z$ by (2). With this the statement follows by Prop. .

In fact the 3-variable adjunction from Prop. even holds internally:

Proposition

(internal tensor/hom-adjunction)

In a symmetric closed monoidal category (def. ) there are natural isomorphisms

$$[X \otimes Y, Z] \;\simeq\; [X, [Y,Z]]$$

whose image under $Hom_{\mathcal{C}}(1,-)$ (see also Example below) are the defining natural bijections of Prop. .

Proof

Let $A \in \mathcal{C}$ be any object. By applying the natural bijections from Prop. , there are composite natural bijections

$$\begin{aligned} Hom_{\mathcal{C}}(A , [X \otimes Y, Z]) & \simeq Hom_{\mathcal{C}}(A \otimes (X \otimes Y), Z) \\ & \simeq Hom_{\mathcal{C}}((A \otimes X)\otimes Y, Z) \\ & \simeq Hom_{\mathcal{C}}(A \otimes X, [Y,Z]) \\ & \simeq Hom_{\mathcal{C}}(A, [X, [Y,Z]]) \end{aligned}$$

Since this holds for all $A$, the fully faithfulness of the Yoneda embedding (Prop. ) says that there is an isomorphism $[ X\otimes Y, Z ] \simeq [X, [Y,Z]]$. Moreover, by taking $A = 1$ in the above and using the left unitor isomorphisms $A \otimes (X \otimes Y) \simeq X \otimes Y$ and $A\otimes X \simeq X$ we get a commuting diagram

$$\array{ Hom_{\mathcal{C}}(1, [X\otimes Y, Z )) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(1, [X, [Y,Z]]) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ Hom_{\mathcal{C}}(X \otimes Y, Z) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(X, [Y,Z]) } \,.$$

Also the key respect of the hom-functor for limits is inherited by internal hom-functors

Proposition

(internal hom preserves limits)

Let $\mathcal{C}$ be a symmetric closed monoidal category with internal hom-bifunctor $[-,-]$ (Prop. ). Then this bifunctor preserves limits in the second variable, and sends colimits in the first variable to limits:

$$[X, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Y(j)] \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [X, Y(j)]$$

and

$$[\underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j),X] \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j),X]$$

Proof

For $X \in \mathcal{X}$ any object, $[X,-]$ is a right adjoint by definition, and hence preserves limits by Prop. .

For the other case, let $Y \;\colon\; \mathcal{L} \to \mathcal{C}$ be a diagram in $\mathcal{C}$, and let $C \in \mathcal{C}$ be any object. Then there are isomorphisms

$$\begin{aligned} Hom_{\mathcal{C}}(C, \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ) & \simeq Hom_{\mathcal{C}}( C \otimes \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ) \\ & \simeq Hom_{\mathcal{C}}( \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} (C \otimes Y(j)), X ) \\ & \simeq \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Hom_{\mathcal{C}}( (C \otimes Y(j)), X ) \\ & \simeq \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Hom_{\mathcal{C}}( C, [Y(j), X] ) \\ & \simeq Hom_{\mathcal{C}}( C, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] ) \end{aligned}$$

which are natural in $C \in \mathcal{C}$, where we used that the ordinary hom-functor preserves limits (Prop. ), and that the left adjoint $C \otimes (-)$ preserves colimits, since left adjoints preserve colimits (Prop. ).

Hence by the fully faithfulness of the Yoneda embedding, there is an isomorphism

$$\left[ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X \right] \overset{\simeq}{\longrightarrow} \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] \,.$$

$\,$

Now that we have seen monoidal categories with various extra properties, we next look at functors which preserve these:

Definition

(monoidal functors)

Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two monoidal categories (def. ). A lax monoidal functor between them is

1. a functor (Def. )

   $$F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,,$$

2. a morphism

   (7)$$\epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})$$

3. a natural transformation (Def. )

   (8)$$\mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y)$$

   for all $x,y \in \mathcal{C}$

satisfying the following conditions:

1. (associativity) For all objects $x,y,z \in \mathcal{C}$ the following diagram commutes

   $$\array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \mu_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z) && F(x) \otimes_{\mathcal{D}} ( F(x \otimes_{\mathcal{C}} y) ) \\ {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z }}} \\ F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) } \,,$$

   where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the associators of the monoidal categories;

2. (unitality) For all $x \in \mathcal{C}$ the following diagrams commutes

   $$\array{ 1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x) &\overset{\epsilon \otimes id}{\longrightarrow}& F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}} \\ F(x) &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}& F(1 \otimes_{\mathcal{C}} x ) }$$

   and

   $$\array{ F(x) \otimes_{\mathcal{D}} 1_{\mathcal{D}} &\overset{id \otimes \epsilon }{\longrightarrow}& F(x) \otimes_{\mathcal{D}} F(1_{\mathcal{C}}) \\ {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}} \\ F(x) &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}& F(x \otimes_{\mathcal{C}} 1 ) } \,,$$

   where $\ell^{\mathcal{C}}$, $\ell^{\mathcal{D}}$, $r^{\mathcal{C}}$, $r^{\mathcal{D}}$ denote the left and right unitors of the two monoidal categories, respectively.

If $\epsilon$ and alll $\mu_{x,y}$ are isomorphisms, then $F$ is called a strong monoidal functor.

If moreover $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are equipped with the structure of braided monoidal categories (def. ) with braidings $\tau^{\mathcal{C}}$ and $\tau^{\mathcal{D}}$, respectively, then the lax monoidal functor $F$ is called a braided monoidal functor if in addition the following diagram commutes for all objects $x,y \in \mathcal{C}$

$$\array{ F(x) \otimes_{\mathcal{C}} F(y) &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}& F(y) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\mu_{x,y}}}\downarrow && \downarrow^{\mathrlap{\mu_{y,x}}} \\ F(x \otimes_{\mathcal{C}} y ) &\underset{F(\tau^{\mathcal{C}}_{x,y} )}{\longrightarrow}& F( y \otimes_{\mathcal{C}} x ) } \,.$$

A homomorphism $f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2)$ between two (braided) lax monoidal functors is a monoidal natural transformation, in that it is a natural transformation $f_x \;\colon\; F_1(x) \longrightarrow F_2(x)$ of the underlying functors

compatible with the product and the unit in that the following diagrams commute for all objects $x,y \in \mathcal{C}$:

$$\array{ F_1(x) \otimes_{\mathcal{D}} F_1(y) &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}& F_2(x) \otimes_{\mathcal{D}} F_2(y) \\ {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\mu_2)_{x,y}}} \\ F_1(x\otimes_{\mathcal{C}} y) &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}& F_2(x \otimes_{\mathcal{C}} y) }$$

and

$$\array{ && 1_{\mathcal{D}} \\ & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}} \\ F_1(1_{\mathcal{C}}) &&\underset{f(1_{\mathcal{C}})}{\longrightarrow}&& F_2(1_{\mathcal{C}}) } \,.$$

We write $MonFun(\mathcal{C},\mathcal{D})$ for the resulting category of lax monoidal functors between monoidal categories $\mathcal{C}$ and $\mathcal{D}$, similarly $BraidMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between braided monoidal categories, and $SymMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between symmetric monoidal categories.

Remark

In the literature the term “monoidal functor” often refers by default to what in def. is called a strong monoidal functor. But for the purpose of the discussion of functors with smash product below, it is crucial to admit the generality of lax monoidal functors.

If $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are symmetric monoidal categories (def. ) then a braided monoidal functor (def. ) between them is often called a symmetric monoidal functor.

Proposition

For $\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D} \overset{G}{\longrightarrow} \mathcal{E}$ two composable lax monoidal functors (def. ) between monoidal categories, then their composite $F \circ G$ becomes a lax monoidal functor with structure morphisms

$$\epsilon^{G\circ F} \;\colon\; 1_{\mathcal{E}} \overset{\epsilon^G}{\longrightarrow} G(1_{\mathcal{D}}) \overset{G(\epsilon^F)}{\longrightarrow} G(F(1_{\mathcal{C}}))$$

and

$$\mu^{G \circ F}_{c_1,c_2} \;\colon\; G(F(c_1)) \otimes_{\mathcal{E}} G(F(c_2)) \overset{\mu^{G}_{F(c_1), F(c_2)}}{\longrightarrow} G( F(c_1) \otimes_{\mathcal{D}} F(c_2) ) \overset{G(\mu^F_{c_1,c_2})}{\longrightarrow} G(F( c_1 \otimes_{\mathcal{C}} c_2 )) \,.$$

Algebras and modules

Definition

Given a monoidal category $(\mathcal{C}, \otimes, 1)$ (Def. ), then a monoid internal to $(\mathcal{C}, \otimes, 1)$ is

1. an object $A \in \mathcal{C}$;

2. a morphism $e \;\colon\; 1 \longrightarrow A$ (called the unit)

3. a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the product);

such that

1. (associativity) the following diagram commutes

   $$\array{ (A\otimes A) \otimes A &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}& A \otimes (A \otimes A) &\overset{A \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes A}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,,$$

   where $a$ is the associator isomorphism of $\mathcal{C}$;

2. (unitality) the following diagram commutes:

   $$\array{ 1 \otimes A &\overset{e \otimes id}{\longrightarrow}& A \otimes A &\overset{id \otimes e}{\longleftarrow}& A \otimes 1 \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\mu}} & & \swarrow_{\mathrlap{r}} \\ && A } \,,$$

   where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.

Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a symmetric monoidal category (def. ) $(\mathcal{C}, \otimes, 1, B)$ with symmetric braiding $\tau$, then a monoid $(A,\mu, e)$ as above is called a commutative monoid in $(\mathcal{C}, \otimes, 1, B)$ if in addition

• (commutativity) the following diagram commutes

  $$\array{ A \otimes A && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} && A \otimes A \\ & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}} \\ && A } \,.$$

A homomorphism of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism

$$f \;\colon\; A_1 \longrightarrow A_2$$

in $\mathcal{C}$, such that the following two diagrams commute

$$\array{ A_1 \otimes A_1 &\overset{f \otimes f}{\longrightarrow}& A_2 \otimes A_2 \\ {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}} \\ A_1 &\underset{f}{\longrightarrow}& A_2 }$$

and

$$\array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,.$$

Write $Mon(\mathcal{C}, \otimes,1)$ for the category of monoids in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its full subcategory of commutative monoids.

Example

Given a monoidal category $(\mathcal{C}, \otimes, 1)$ (Def. ), the tensor unit $1$ is a monoid in $\mathcal{C}$ (def. ) with product given by either the left or right unitor

$$\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1 \,.$$

By lemma , these two morphisms coincide and define an associative product with unit the identity $id \colon 1 \to 1$.

If $(\mathcal{C}, \otimes , 1)$ is a symmetric monoidal category (def. ), then this monoid is a commutative monoid.

Example

Given a symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. ), and given two commutative monoids $(E_i, \mu_i, e_i)$ $i \in \{1,2\}$ (def. ), then the tensor product $E_1 \otimes E_2$ becomes itself a commutative monoid with unit morphism

$$e \;\colon\; 1 \overset{\simeq}{\longrightarrow} 1 \otimes 1 \overset{e_1 \otimes e_2}{\longrightarrow} E_1 \otimes E_2$$

(where the first isomorphism is, $\ell_1^{-1} = r_1^{-1}$ (lemma )) and with product morphism given by

$$E_1 \otimes E_2 \otimes E_1 \otimes E_2 \overset{id \otimes \tau_{E_2, E_1} \otimes id}{\longrightarrow} E_1 \otimes E_1 \otimes E_2 \otimes E_2 \overset{\mu_1 \otimes \mu_2}{\longrightarrow} E_1 \otimes E_2$$

(where we are notationally suppressing the associators and where $\tau$ denotes the braiding of $\mathcal{C}$).

That this definition indeed satisfies associativity and commutativity follows from the corresponding properties of $(E_i,\mu_i, e_i)$, and from the hexagon identities for the braiding (def. ) and from symmetry of the braiding.

Similarly one checks that for $E_1 = E_2 = E$ then the unit maps

$$E \simeq E \otimes 1 \overset{id \otimes e}{\longrightarrow} E \otimes E$$

$$E \simeq 1 \otimes E \overset{e \otimes 1}{\longrightarrow} E \otimes E$$

and the product map

$$\mu \;\colon\; E \otimes E \longrightarrow E$$

and the braiding

$$\tau_{E,E} \;\colon\; E \otimes E \longrightarrow E \otimes E$$

are monoid homomorphisms, with $E \otimes E$ equipped with the above monoid structure.

Definition

Given a monoidal category $(\mathcal{C}, \otimes, 1)$ (def. ), and given $(A,\mu,e)$ a monoid in $(\mathcal{C}, \otimes, 1)$ (def. ), then a left module object in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is

1. an object $N \in \mathcal{C}$;

2. a morphism $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the action);

such that

1. (unitality) the following diagram commutes:

   $$\array{ 1 \otimes N &\overset{e \otimes id}{\longrightarrow}& A \otimes N \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\rho}} \\ && N } \,,$$

   where $\ell$ is the left unitor isomorphism of $\mathcal{C}$.

2. (action property) the following diagram commutes

   $$\array{ (A\otimes A) \otimes N &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}& A \otimes (A \otimes N) &\overset{A \otimes \rho}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \otimes N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,,$$

A homomorphism of left $A$-module objects

$$(N_1, \rho_1) \longrightarrow (N_2, \rho_2)$$

is a morphism

$$f\;\colon\; N_1 \longrightarrow N_2$$

in $\mathcal{C}$, such that the following diagram commutes:

$$\array{ A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2 \\ {}^{\mathllap{\rho_1}}\downarrow && \downarrow^{\mathrlap{\rho_2}} \\ N_1 &\underset{f}{\longrightarrow}& N_2 } \,.$$

For the resulting category of modules of left $A$-modules in $\mathcal{C}$ with $A$-module homomorphisms between them, we write

$$A Mod(\mathcal{C}) \,.$$

Example

Given a monoidal category $(\mathcal{C},\otimes, 1)$ (def. ) with the tensor unit $1$ regarded as a monoid in a monoidal category via example , then the left unitor

$$\ell_C \;\colon\; 1\otimes C \longrightarrow C$$

makes every object $C \in \mathcal{C}$ into a left module, according to def. , over $C$. The action property holds due to lemma . This gives an equivalence of categories

$$\mathcal{C} \simeq 1 Mod(\mathcal{C})$$

of $\mathcal{C}$ with the category of modules over its tensor unit.

Example

The archetypical case in which all these abstract concepts reduce to the basic familiar ones is the symmetric monoidal category Ab of abelian groups from example .

1. A monoid in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently a ring.

2. A commutative monoid in in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently a commutative ring $R$.

3. An $R$-module object in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently an $R$-module;

4. The tensor product of $R$-module objects (def. ) is the standard tensor product of modules.

5. The category of module objects $R Mod(Ab)$ (def. ) is the standard category of modules $R Mod$.

Example

Closely related to the example , but closer to the structure we will see below for spectra, are monoids in the category of chain complexes $(Ch_\bullet, \otimes, \mathbb{Z})$ from example . These monoids are equivalently differential graded algebras.

Proposition

In the situation of def. , the monoid $(A,\mu, e)$ canonically becomes a left module over itself by setting $\rho \coloneqq \mu$. More generally, for $C \in \mathcal{C}$ any object, then $A \otimes C$ naturally becomes a left $A$-module by setting:

$$\rho \;\colon\; A \otimes (A \otimes C) \underoverset{\simeq}{a^{-1}_{A,A,C}}{\longrightarrow} (A \otimes A) \otimes C \overset{\mu \otimes id}{\longrightarrow} A \otimes C \,.$$

The $A$-modules of this form are called free modules.

The free functor $F$ constructing free $A$-modules is left adjoint to the forgetful functor $U$ which sends a module $(N,\rho)$ to the underlying object $U(N,\rho) \coloneqq N$.

$$A Mod(\mathcal{C}) \underoverset {\underset{U}{\longrightarrow}} {\overset{F}{\longleftarrow}} {\bot} \mathcal{C} \,.$$

Proof

A homomorphism out of a free $A$-module is a morphism in $\mathcal{C}$ of the form

$$f \;\colon\; A\otimes C \longrightarrow N$$

fitting into the diagram (where we are notationally suppressing the associator)

$$\array{ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,.$$

Consider the composite

$$\tilde f \;\colon\; C \underoverset{\simeq}{\ell_C}{\longrightarrow} 1 \otimes C \overset{e\otimes id}{\longrightarrow} A \otimes C \overset{f}{\longrightarrow} N \,,$$

i.e. the restriction of $f$ to the unit “in” $A$. By definition, this fits into a commuting square of the form (where we are now notationally suppressing the associator and the unitor)

$$\array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\underset{id \otimes f}{\longrightarrow}& A \otimes N } \,.$$

Pasting this square onto the top of the previous one yields

$$\array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,,$$

where now the left vertical composite is the identity, by the unit law in $A$. This shows that $f$ is uniquely determined by $\tilde f$ via the relation

$$f = \rho \circ (id_A \otimes \tilde f) \,.$$

This natural bijection between $f$ and $\tilde f$ establishes the adjunction.

Definition

Given a closed symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. , def. ), given $(A,\mu,e)$ a commutative monoid in $(\mathcal{C}, \otimes, 1)$ (def. ), and given $(N_1, \rho_1)$ and $(N_2, \rho_2)$ two left $A$-module objects (def.), then

1. the tensor product of modules $N_1 \otimes_A N_2$ is, if it exists, the coequalizer

   $$N_1 \otimes A \otimes N_2 \underoverset {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}} {\overset{N_1 \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} N_1 \otimes N_1 \overset{coeq}{\longrightarrow} N_1 \otimes_A N_2$$

   and if $A \otimes (-)$ preserves these coequalizers, then this is equipped with the left $A$-action induced from the left $A$-action on $N_1$

2. the function module $hom_A(N_1,N_2)$ is, if it exists, the equalizer

   $$hom_A(N_1, N_2) \overset{equ}{\longrightarrow} hom(N_1, N_2) \underoverset {\underset{hom(A \otimes N_1, \rho_2)\circ (A \otimes(-))}{\longrightarrow}} {\overset{hom(\rho_1,N_2)}{\longrightarrow}} {\phantom{AAAAAA}} hom(A \otimes N_1, N_2) \,.$$

   equipped with the left $A$-action that is induced by the left $A$-action on $N_2$ via

   $$\frac{ A \otimes hom(X,N_2) \longrightarrow hom(X,N_2) }{ A \otimes hom(X,N_2) \otimes X \overset{id \otimes ev}{\longrightarrow} A \otimes N_2 \overset{\rho_2}{\longrightarrow} N_2 } \,.$$

(e.g. Hovey-Shipley-Smith 00, lemma 2.2.2 and lemma 2.2.8)

Proposition

Given a closed symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. , def. ), and given $(A,\mu,e)$ a commutative monoid in $(\mathcal{C}, \otimes, 1)$ (def. ). If all coequalizers exist in $\mathcal{C}$, then the tensor product of modules $\otimes_A$ from def. makes the category of modules $A Mod(\mathcal{C})$ into a symmetric monoidal category, $(A Mod, \otimes_A, A)$ with tensor unit the object $A$ itself, regarded as an $A$-module via prop. .

If moreover all equalizers exist, then this is a closed monoidal category (def. ) with internal hom given by the function modules $hom_A$ of def. .

(e.g. Hovey-Shipley-Smith 00, lemma 2.2.2, lemma 2.2.8)

Proof sketch

The associators and braiding for $\otimes_{A}$ are induced directly from those of $\otimes$ and the universal property of coequalizers. That $A$ is the tensor unit for $\otimes_{A}$ follows with the same kind of argument that we give in the proof of example below.

Example

For $(A,\mu,e)$ a monoid (def. ) in a symmetric monoidal category $(\mathcal{C},\otimes, 1)$ (def. ), the tensor product of modules (def. ) of two free modules (def. ) $A\otimes C_1$ and $A \otimes C_2$ always exists and is the free module over the tensor product in $\mathcal{C}$ of the two generators:

$$(A \otimes C_1) \otimes_A (A \otimes C_2) \simeq A \otimes (C_1 \otimes C_2) \,.$$

Hence if $\mathcal{C}$ has all coequalizers, so that the category of modules is a monoidal category $(A Mod, \otimes_A, A)$ (prop. ) then the free module functor (def. ) is a strong monoidal functor (def. )

$$F \;\colon\; (\mathcal{C}, \otimes, 1) \longrightarrow (A Mod, \otimes_A, A) \,.$$

Proof

It is sufficient to show that the diagram

$$A \otimes A \otimes A \underoverset {\underset{id \otimes \mu}{\longrightarrow}} {\overset{\mu \otimes id}{\longrightarrow}} {\phantom{AAAA}} A \otimes A \overset{\mu}{\longrightarrow} A$$

is a coequalizer diagram (we are notationally suppressing the associators), hence that $A \otimes_A A \simeq A$, hence that the claim holds for $C_1 = 1$ and $C_2 = 1$.

To that end, we check the universal property of the coequalizer:

First observe that $\mu$ indeed coequalizes $id \otimes \mu$ with $\mu \otimes id$, since this is just the associativity clause in def. . So for $f \colon A \otimes A \longrightarrow Q$ any other morphism with this property, we need to show that there is a unique morphism $\phi \colon A \longrightarrow Q$ which makes this diagram commute:

$$\array{ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{\phi}} \\ Q } \,.$$

We claim that

$$\phi \;\colon\; A \underoverset{\simeq}{r^{-1}}{\longrightarrow} A \otimes 1 \overset{id \otimes e}{\longrightarrow} A \otimes A \overset{f}{\longrightarrow} Q \,,$$

where the first morphism is the inverse of the right unitor of $\mathcal{C}$.

First to see that this does make the required triangle commute, consider the following pasting composite of commuting diagrams

$$\array{ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{id \otimes r^{-1}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{r^{-1}}}_{\simeq} \\ A \otimes A \otimes 1 &\overset{\mu \otimes id}{\longrightarrow}& A \otimes 1 \\ {}^{\mathllap{id \otimes e}}\downarrow && \downarrow^{\mathrlap{id \otimes e} } \\ A \otimes A \otimes A &\overset{\mu \otimes id}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{id \otimes \mu}}\downarrow && \downarrow^{\mathrlap{f}} \\ A \otimes A &\underset{f}{\longrightarrow}& Q } \,.$$

Here the the top square is the naturality of the right unitor, the middle square commutes by the functoriality of the tensor product $\otimes \;\colon\; \mathcal{C}\times \mathcal{C} \longrightarrow \mathcal{C}$ and the definition of the product category (Example ), while the commutativity of the bottom square is the assumption that $f$ coequalizes $id \otimes \mu$ with $\mu \otimes id$.

Here the right vertical composite is $\phi$, while, by unitality of $(A,\mu ,e)$, the left vertical composite is the identity on $A$, Hence the diagram says that $\phi \circ \mu = f$, which we needed to show.

It remains to see that $\phi$ is the unique morphism with this property for given $f$. For that let $q \colon A \to Q$ be any other morphism with $q\circ \mu = f$. Then consider the commuting diagram

$$\array{ A \otimes 1 &\overset{\simeq}{\longleftarrow}& A \\ {}^{\mathllap{id\otimes e}}\downarrow & \searrow^{\simeq} & \downarrow^{\mathrlap{=}} \\ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{q}} \\ Q } \,,$$

where the top left triangle is the unitality condition and the two isomorphisms are the right unitor and its inverse. The commutativity of this diagram says that $q = \phi$.

Definition

Given a monoidal category of modules $(A Mod , \otimes_A , A)$ as in prop. , then a monoid $(E, \mu, e)$ in $(A Mod , \otimes_A , A)$ (def. ) is called an $A$-algebra.

Proposition

Given a monoidal category of modules $(A Mod , \otimes_A , A)$ in a monoidal category $(\mathcal{C},\otimes, 1)$ as in prop. , and an $A$-algebra $(E,\mu,e)$ (def. ), then there is an equivalence of categories

$$A Alg_{comm}(\mathcal{C}) \coloneqq CMon(A Mod) \simeq CMon(\mathcal{C})^{A/}$$

between the category of commutative monoids in $A Mod$ and the coslice category of commutative monoids in $\mathcal{C}$ under $A$, hence between commutative $A$-algebras in $\mathcal{C}$ and commutative monoids $E$ in $\mathcal{C}$ that are equipped with a homomorphism of monoids $A \longrightarrow E$.

(e.g. EKMM 97, VII lemma 1.3)

Proof

In one direction, consider a $A$-algebra $E$ with unit $e_E \;\colon\; A \longrightarrow E$ and product $\mu_{E/A} \colon E \otimes_A E \longrightarrow E$. There is the underlying product $\mu_E$

$$\array{ E \otimes A \otimes E & \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longrightarrow}} {\phantom{AAA}} & E \otimes E &\overset{coeq}{\longrightarrow}& E \otimes_A E \\ && & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && && E } \,.$$

By considering a diagram of such coequalizer diagrams with middle vertical morphism $e_E\circ e_A$, one find that this is a unit for $\mu_E$ and that $(E, \mu_E, e_E \circ e_A)$ is a commutative monoid in $(\mathcal{C}, \otimes, 1)$.

Then consider the two conditions on the unit $e_E \colon A \longrightarrow E$. First of all this is an $A$-module homomorphism, which means that

$$(\star) \;\;\;\;\; \;\;\;\;\; \array{ A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A &\underset{e_E}{\longrightarrow}& E }$$

commutes. Moreover it satisfies the unit property

$$\array{ A \otimes_A E &\overset{e_A \otimes id}{\longrightarrow}& E \otimes_A E \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && E } \,.$$

By forgetting the tensor product over $A$, the latter gives

$$\array{ A \otimes E &\overset{e \otimes id}{\longrightarrow}& E \otimes E \\ \downarrow && \downarrow^{\mathrlap{}} \\ A \otimes_A E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes_A E \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_{E/A}}} \\ E &=& E } \;\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;\; \array{ A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ E &\underset{id}{\longrightarrow}& E } \,,$$

where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be pasted to the square $(\star)$ above, to yield a commuting square

$$\array{ A \otimes A &\overset{id\otimes e_E}{\longrightarrow}& A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \array{ A \otimes A &\overset{e_E \otimes e_E}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\mu_E}} \\ A &\underset{e_E}{\longrightarrow}& E } \,.$$

This shows that the unit $e_A$ is a homomorphism of monoids $(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A)$.

Now for the converse direction, assume that $(A,\mu_A, e_A)$ and $(E, \mu_E, e'_E)$ are two commutative monoids in $(\mathcal{C}, \otimes, 1)$ with $e_E \;\colon\; A \to E$ a monoid homomorphism. Then $E$ inherits a left $A$-module structure by

$$\rho \;\colon\; A \otimes E \overset{e_A \otimes id}{\longrightarrow} E \otimes E \overset{\mu_E}{\longrightarrow} E \,.$$

By commutativity and associativity it follows that $\mu_E$ coequalizes the two induced morphisms $E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E$. Hence the universal property of the coequalizer gives a factorization through some $\mu_{E/A}\colon E \otimes_A E \longrightarrow E$. This shows that $(E, \mu_{E/A}, e_E)$ is a commutative $A$-algebra.

Finally one checks that these two constructions are inverses to each other, up to isomorphism.

Definition

(lax monoidal functor)

Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two monoidal categories (def. ). A lax monoidal functor between them is

1. a functor

   $$F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,,$$

2. a morphism

   $$\epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})$$

3. a natural transformation

   $$\mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y)$$

   for all $x,y \in \mathcal{C}$

satisfying the following conditions:

1. (associativity) For all objects $x,y,z \in \mathcal{C}$ the following diagram commutes

   $$\array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \mu_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z) && F(x) \otimes_{\mathcal{D}} ( F(x \otimes_{\mathcal{C}} y) ) \\ {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z }}} \\ F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) } \,,$$

   where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the associators of the monoidal categories;

2. (unitality) For all $x \in \mathcal{C}$ the following diagrams commutes

   $$\array{ 1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x) &\overset{\epsilon \otimes id}{\longrightarrow}& F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}} \\ F(x) &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}& F(1 \otimes_{\mathcal{C}} x ) }$$

   and

   $$\array{ F(x) \otimes_{\mathcal{D}} 1_{\mathcal{D}} &\overset{id \otimes \epsilon }{\longrightarrow}& F(x) \otimes_{\mathcal{D}} F(1_{\mathcal{C}}) \\ {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}} \\ F(x) &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}& F(x \otimes_{\mathcal{C}} 1 ) } \,,$$

   where $\ell^{\mathcal{C}}$, $\ell^{\mathcal{D}}$, $r^{\mathcal{C}}$, $r^{\mathcal{D}}$ denote the left and right unitors of the two monoidal categories, respectively.

If $\epsilon$ and alll $\mu_{x,y}$ are isomorphisms, then $F$ is called a strong monoidal functor.

If moreover $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are equipped with the structure of braided monoidal categories (def. ) with braidings $\tau^{\mathcal{C}}$ and $\tau^{\mathcal{D}}$, respectively, then the lax monoidal functor $F$ is called a braided monoidal functor if in addition the following diagram commutes for all objects $x,y \in \mathcal{C}$

$$\array{ F(x) \otimes_{\mathcal{C}} F(y) &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}& F(y) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\mu_{x,y}}}\downarrow && \downarrow^{\mathrlap{\mu_{y,x}}} \\ F(x \otimes_{\mathcal{C}} y ) &\underset{F(\tau^{\mathcal{C}}_{x,y} )}{\longrightarrow}& F( y \otimes_{\mathcal{C}} x ) } \,.$$

A homomorphism $f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2)$ between two (braided) lax monoidal functors is a monoidal natural transformation, in that it is a natural transformation $f_x \;\colon\; F_1(x) \longrightarrow F_2(x)$ of the underlying functors

compatible with the product and the unit in that the following diagrams commute for all objects $x,y \in \mathcal{C}$:

$$\array{ F_1(x) \otimes_{\mathcal{D}} F_1(y) &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}& F_2(x) \otimes_{\mathcal{D}} F_2(y) \\ {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\mu_2)_{x,y}}} \\ F_1(x\otimes_{\mathcal{C}} y) &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}& F_2(x \otimes_{\mathcal{C}} y) }$$

and

$$\array{ && 1_{\mathcal{D}} \\ & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}} \\ F_1(1_{\mathcal{C}}) &&\underset{f(1_{\mathcal{C}})}{\longrightarrow}&& F_2(1_{\mathcal{C}}) } \,.$$

We write $MonFun(\mathcal{C},\mathcal{D})$ for the resulting category of lax monoidal functors between monoidal categories $\mathcal{C}$ and $\mathcal{D}$, similarly $BraidMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between braided monoidal categories, and $SymMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between symmetric monoidal categories.

Remark

In the literature the term “monoidal functor” often refers by default to what in def. is called a strong monoidal functor. But for the purpose of the discussion of functors with smash product below, it is crucial to admit the generality of lax monoidal functors.

If $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are symmetric monoidal categories (def. ) then a braided monoidal functor (def. ) between them is often called a symmetric monoidal functor.

Proposition

For $\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D} \overset{G}{\longrightarrow} \mathcal{E}$ two composable lax monoidal functors (def. ) between monoidal categories, then their composite $F \circ G$ becomes a lax monoidal functor with structure morphisms

$$\epsilon^{G\circ F} \;\colon\; 1_{\mathcal{E}} \overset{\epsilon^G}{\longrightarrow} G(1_{\mathcal{D}}) \overset{G(\epsilon^F)}{\longrightarrow} G(F(1_{\mathcal{C}}))$$

and

$$\mu^{G \circ F}_{c_1,c_2} \;\colon\; G(F(c_1)) \otimes_{\mathcal{E}} G(F(c_2)) \overset{\mu^{G}_{F(c_1), F(c_2)}}{\longrightarrow} G( F(c_1) \otimes_{\mathcal{D}} F(c_2) ) \overset{G(\mu^F_{c_1,c_2})}{\longrightarrow} G(F( c_1 \otimes_{\mathcal{C}} c_2 )) \,.$$

Proposition

(lax monoidal functors preserve monoids)

Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D}, \otimes_{\mathcal{D}},1_{\mathcal{D}})$ be two monoidal categories (def. ) and let $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a lax monoidal functor (def. ) between them.

Then for $(A,\mu_A,e_A)$ a monoid in $\mathcal{C}$ (def. ), its image $F(A) \in \mathcal{D}$ becomes a monoid $(F(A), \mu_{F(A)}, e_{F(A)})$ by setting

$$\mu_{F(A)} \;\colon\; F(A) \otimes_{\mathcal{C}} F(A) \overset{}{\longrightarrow} F(A \otimes_{\mathcal{C}} A) \overset{F(\mu_A)}{\longrightarrow} F(A)$$

(where the first morphism is the structure morphism of $F$) and setting

$$e_{F(A)} \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}}) \overset{F(e_A)}{\longrightarrow} F(A)$$

(where again the first morphism is the corresponding structure morphism of $F$).

This construction extends to a functor

$$Mon(F) \;\colon\; Mon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}}) \longrightarrow Mon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}})$$

from the category of monoids of $\mathcal{C}$ (def. ) to that of $\mathcal{D}$.

Moreover, if $\mathcal{C}$ and $\mathcal{D}$ are symmetric monoidal categories (def. ) and $F$ is a braided monoidal functor (def. ) and $A$ is a commutative monoid (def. ) then so is $F(A)$, and this construction extends to a functor between categories of commutative monoids:

$$CMon(F) \;\colon\; CMon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}}) \longrightarrow CMon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}}) \,.$$

Proof

This follows immediately from combining the associativity and unitality (and symmetry) constraints of $F$ with those of $A$.

$\,$

Enriched categories

The plain definition of categories in Def. is phrased in terms of sets. Via Example this assigns a special role to the category Set of all sets, as the “base” on top, or the “cosmos” inside which category theory takes place. For instance, the fact that hom-sets in a plain category are indeed sets, is what makes the hom-functor (Example ) take values in Set, and this, in turn, governs the form of the all-important Yoneda lemma (Prop. ) and Yoneda embedding (Prop. ) as statements about presheaves of sets (Example ).

At the same time, category theory witnesses the utility of abstracting away from concrete choices to their abstract properties that are actually used in constructions. This makes it natural to ask if one could replace the category Set by some other category $\mathcal{V}$ which could similarly serve as a “cosmos” inside which category theory may be developed.

Indeed, such $\mathcal{V}$-enriched category theory (see Example below for the terminology) exists, beginning with the concept of $\mathcal{V}$-enriched categories (Def. below) and from there directly paralleling, hence generalizing, plain category theory, as long as one assumes the “cosmos” category $\mathcal{V}$ to share a minimum of abstract properties with Set (Def. below).

This turns out to be most useful. In fact, the perspective of enriched categories is helpful already when $\mathcal{V} =$ Set, in which case it reproduces plain category theory (Example below), for instance in that it puts the (co)limits of the special form of (co)ends (Def. below) to the forefront (discussed below).

$\,$

Definition

(cosmos)

A Bénabou cosmos for enriched category theory, or just cosmos, for short, is a symmetric (Def. ) closed monoidal category (Def. ) $\mathcal{V}$ which has all limits and colimits.

Example

(examples of cosmoi for enriched category theory)

The following are examples of cosmoi (Def. ):

1. $Sh(\mathcal{C})$ the sheaf topos (Def. ) over any site (Def. ) – by Prop. below.

   In particular:

   1. Set (Def. ) equipped with its cartesian closed category-structure (Example )

   2. sSet$\simeq [\Delta^{op}, Set]$ (Def. , Prop. )

2. Grpd (Def. ) equipped with its cartesian closed category-structure (Example ).

3. Cat (Def. ) equipped with its cartesian closed category-structure (Example ).

Example

underlying set of an object in a cosmos

Let $\mathcal{V}$ be a cosmos (Def. ), with $1 \in \mathcal{V}$ its tensor unit (Def. ). Then the hom-functor (Def. ) out of $1$

$$Hom_{\mathcal{V}}(1,-) \;\colon\; \mathcal{V} \longrightarrow Set$$

admits the structure of a lax monoidal functor (Def. ) to Set, with the latter regarded with its cartesian monoidal structure from Example .

Given $V \in \mathcal{V}$, we call

$$Hom_{\mathcal{V}}(1,V) \;\in\; Set$$

also the underlying set of $V$.

Proof

Take the monoidal transformations (eq“MonoidalComponentsOfMonoidalFunctor) to be

$$\array{ Hom_{\mathcal{V}}(1, V_1) \times Hom_{\mathcal{V}}(1, V_2) \longrightarrow Hom_{\mathcal{V}}(1, V_1 \otimes V_2) \\ \big( 1 \overset{f_1}{\to} V_1\;,\; 1 \overset{f_2}{\to} V_2 \big) &\mapsto& \big( 1 \overset{\simeq}{\to} 1 \otimes 1 \overset{f_1 \otimes f_2}{\longrightarrow} V_1 \otimes V_2 \big) }$$

and take the unit transformation (7)

$$\array{ \ast &\longrightarrow& Hom_{\mathcal{V}}(1,1) }$$

to pick $id_1 \in Hom_{\mathcal{V}}(1,1)$.

Example

(underlying set of internal hom is hom-set)*

For $\mathcal{V}$ a cosmos (Def. ), let $X,Y \in Obj_{\mathcal{V}}$ be two objects. Then the underlying set (Def. ) of their internal hom $[X,Y] \in \mathcal{V}$ (Def. ) is the hom-set (Def. ):

$$\mathcal{Hom}_{\mathcal{V}}\left( 1, [X,Y]\right) \;\simeq\; Hom_{\mathcal{V}}(X,Y) \,.$$

This identification is the adjunction isomorphism (?) for the internal hom adjunction (2) followed composed with a unitor (Def. ).

Definition

(enriched category)

For $\mathcal{V}$ a cosmos (Def. ), a $\mathcal{V}$-enriched category $\mathcal{C}$ is:

1. a class $Obj_{\mathcal{C}}$, called the class of objects;

2. for each $a, b\in Obj_{\mathcal{C}}$, an object

   $$\mathcal{C}(a,b)\in \mathcal{V} \,,$$

   called the $\mathcal{V}$-object of morphisms between $a$ and $b$;

3. for each $a,b,c\in Obj(\mathcal{C})$ a morphism in $\mathcal{V}$

   $$\circ_{a,b,c} \;\colon\; \mathcal{C}(a,b)\times \mathcal{C}(b,c) \longrightarrow \mathcal{C}(a,c)$$

   out of the tensor product of hom-objects, called the composition operation;

4. for each $a \in Obj(\mathcal{C})$ a morphism $Id_a \colon \ast \to \mathcal{C}(a,a)$, called the identity morphism on $a$

such that the composition is associative and unital.

If the class $Obj_{\mathcal{C}}$ happens to be a set (hence a small set instead of a proper class) then we say the $\mathcal{V}$-enriched category $\mathcal{C}$ is small, as in Def. .

Example

(Set-enriched categories are plain categories)

An enriched category (Def. ) over the cosmos $\mathcal{V} =$ Set, as in Example , is the same as a plain category (Def. ).

Example

(Cat-enriched categories are strict 2-categories)

An enriched category (Def. ) over the cosmos $\mathcal{V} =$ Cat, as in Example , is the same as a strict 2-category (Def. ).

Example

(underlying category of an enriched category)

Let $\mathcal{C}$ be a $\mathcal{V}$-enriched category (Def. ).

Using the lax monoidal structure (Def. ) on the hom functor (Example )

$$Hom_{\mathcal{V}}(1,-) \;\colon\; \mathcal{V} \longrightarrow Set$$

out of the tensor unit $1 \in \mathcal{C}$ this induces a Set-enriched category $\vert \mathcal{C}\vert$ with hence an ordinary category (Example ), with

• $Obj_{\vert \mathcal{C}\vert} \;\coloneqq\; Obj_{\mathcal{C}}$;

• $Hom_{\vert \mathcal{C}\vert}( X, Y ) \;\coloneqq\; Hom_{\mathcal{V}}(1, \mathcal{C}(X,Y))$.

It is in this sense that $\mathcal{C}$ is a plain category ${\vert \mathcal{C}\vert}$ equipped with extra structure, and hence an “enriched category”.

The archetypical example is $\mathcal{V}$ itself:

Example

($\mathcal{V}$ as a $\mathcal{V}$-enriched category)

Evert cosmos $\mathcal{C}$ (Def. ) canonically obtains the structure of a $\mathcal{V}$-enriched category, def. :

the hom-objects are the internal homs

$$\mathcal{v}(X,Y) \coloneqq [X,Y]$$

and with composition

$$[X,Y] \times [Y,Z] \longrightarrow [X,Z]$$

given by the adjunct under the (Cartesian product$\dashv$ internal hom)-adjunction of the evaluation morphisms

$$X \otimes [XmY] \otimes [Y,Z] \overset{(ev, id)}{\longrightarrow} Y \otimes [Y,Z] \overset{ev}{\longrightarrow} Z \,.$$

The usual construction on categories, such as that of opposite categories (Def. ) and product categories (Def. ) have evident enriched analogs

Definition

(enriched opposite category and product category)

For $\mathcal{V}$ a cosmos, let $\mathcal{C}, \mathcal{D}$ be $\mathcal{V}$-enriched categories (Def. ).

1. The opposite enriched category $\mathcal{C}^{op}$ is the enriched category with the same objects as $\mathcal{C}$, with hom-objects

   $$\mathcal{C}^{op}(X,Y) \coloneqq \mathcal{C}(Y,X)$$

   and with composition given by braiding (1) followed by the composition in $\mathcal{C}$:

   $$\mathcal{C}^{op}(X,Y) \otimes \mathcal{C}^{op}(Y,Z) = \mathcal{C}(Y,X) \otimes \mathcal{C}(Z,Y) \underoverset{\simeq}{\tau}{\longrightarrow} \mathcal{C}(Z,Y) \otimes \mathcal{C}(Y,X) \overset{\circ_{Z,Y,X}}{\longrightarrow} \mathcal{C}(Z,X) = \mathcal{C}^{op}(X,Z) \,.$$

2. the enriched product category $\mathcal{C} \times \mathcal{D}$ is the enriched category whose objects are pairs of objects $(c,d)$ with $c \in \mathcal{C}$ and $d\in \mathcal{D}$, whose hom-spaces are the tensor product of the separate hom objects

   $$(\mathcal{C}\times \mathcal{D})((c_1,d_1),\;(c_2,d_2) ) \coloneqq \mathcal{C}(c_1,c_2)\otimes \mathcal{D}(d_1,d_2)$$

   and whose composition operation is the braiding (1) followed by the tensor product of the separate composition operations:

   $$\array{ (\mathcal{C}\times \mathcal{D})((c_1,d_1), \; (c_2,d_2)) \otimes (\mathcal{C}\times \mathcal{D})((c_2,d_2), \; (c_3,d_3)) \\ {}^{\mathllap{=}}\downarrow \\ \left(\mathcal{C}(c_1,c_2) \otimes \mathcal{D}(d_1,d_2)\right) \otimes \left(\mathcal{C}(c_2,c_3) \otimes \mathcal{D}(d_2,d_3)\right) \\ \downarrow^{\mathrlap{\tau}}_{\mathrlap{\simeq}} \\ \left(\mathcal{C}(c_1,c_2) \otimes \mathcal{C}(c_2,c_3)\right) \otimes \left( \mathcal{D}(d_1,d_2) \otimes \mathcal{D}(d_2,d_3)\right) &\overset{ (\circ_{c_1,c_2,c_3}) \otimes (\circ_{d_1,d_2,d_3}) }{\longrightarrow} & \mathcal{C}(c_1,c_3) \otimes \mathcal{D}(d_1,d_3) \\ && \downarrow^{\mathrlap{=}} \\ && (\mathcal{C}\times \mathcal{D})((c_1,d_1),\; (c_3,d_3)) } \,.$$

Definition

(enriched functor)

For $\mathcal{V}$ a cosmos (Def. ), let $\mathcal{C}$ and $\mathcal{D}$ be two $\mathcal{V}$-enriched categories (Def. ).

A $\mathcal{V}$-enriched functor from $\mathcal{C}$ to $\mathcal{D}$

$$F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$$

is

1. a function

   $$F_{Obj} \;\colon\; Obj_{\mathcal{C}} \longrightarrow Obj_{\mathcal{D}}$$

   of objects;

2. for each $a,b \in Obj_{\mathcal{C}}$ a morphism in $\mathcal{V}$

   $$F_{a,b} \;\colon\; \mathcal{C}(a,b) \longrightarrow \mathcal{D}(F_0(a), F_0(b))$$

   between hom-objects

such that this preserves composition and identity morphisms in the evident sense.

Example

(enriched hom-functor)

For $\mathcal{V}$ a cosmos (Def. ), let $\mathcal{C}$ be a $\mathcal{V}$-enriched category (Def. ). Then there is a $\mathcal{V}$-enriched functor out of the enriched product category of $\mathcal{C}$ with its enriched opposite category (Def. )

$$\mathcal{C}(-,-) \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{V}$$

to $\mathcal{V}$, regarded as a $\mathcal{V}$-enriched category (Example ), which sends a pair of objects $X,Y \in \mathcal{C}$ to the hom-object $\mathcal{C}(X,Y) \in \mathcal{V}$, and which acts on morphisms by composition in the evident way.

Example

(enriched presheaves)

For $\mathcal{V}$ a cosmos (Def. ), let $\mathcal{C}$ be a $\mathcal{V}$-enriched category (Def. ). Then a $\mathcal{V}$-enriched functor (Def. )

$$F \;\colon\; \mathcal{C} \longrightarrow \mathcal{V}$$

to the archetypical $\mathcal{V}$-enriched category from Example is:

1. an object $F_a \in Obj_{\mathcal{V}}$ for each object $a \in Obj_{\mathcal{C}}$;

2. a morphism in $\mathcal{V}$ of the form

   $$F_a \otimes \mathcal{C}(a,b) \longrightarrow F_b$$

   for all pairs of objects $a,b \in Obj(\mathcal{C})$

   (this is the adjunct of $F_{a,b}$ under the adjunction (2) on $\mathcal{V}$)

such that composition is respected, in the evident sense.

For every object $c \in \mathcal{C}$, there is an enriched representable functor, denoted

$$y(c) \;\coloneqq\; \mathcal{C}(c,-)$$

(where on the right we have the enriched hom-functor from Example )

which sends objects to

$$y(c)(d) = \mathcal{C}(c,d) \in \mathcal{V}$$

and whose action on morphisms is, under the above identification, just the composition operation in $\mathcal{C}$.

More generally, the following situation will be of interest:

Example

(enriched functor on enriched product category with opposite category)

An $\mathcal{V}$-enriched functor (Def. ) into $\mathcal{V}$ (Example ) out of an enriched product category (Def. )

$$F \;\colon\; \mathcal{C} \times \mathcal{D} \longrightarrow \mathcal{V}$$

(an “enriched bifunctor”) has component morphisms of the form

$$F_{(c_1,d_1),(c_2,d_2)} \;\colon\; \mathcal{C}(c_1,c_2) \otimes \mathcal{D}(d_1,d_2) \longrightarrow \big[ F_0((c_1,d_1)),F_0((c_2,d_2)) \big] \,.$$

By functoriality and under passing to adjuncts (Def. ) under (2) this is equivalent to two commuting actions

$$\rho_{c_1,c_2}(d) \;\colon\; \mathcal{C}(c_1,c_2) \otimes F_0((c_1,d)) \longrightarrow F_0((c_2,d))$$

and

$$\rho_{d_1,d_2}(c) \;\colon\; \mathcal{D}(d_1,d_2) \otimes F_0((c,d_1)) \longrightarrow F_0((c,d_2)) \,.$$

In the special case of a functor out of the enriched product category of some $\mathcal{V}$-enriched category $\mathcal{C}$ with its enriched opposite category (def. )

$$F \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{V}$$

then this takes the form of a “pullback action” in the first variable

$$\rho_{c_2,c_1}(d) \;\colon\; \mathcal{C}(c_1,c_2) \otimes F_0((c_2,d)) \longrightarrow F_0((c_1,d))$$

and a “pushforward action” in the second variable

$$\rho_{d_1,d_2}(c) \;\colon\; \mathcal{C}(d_1,d_2) \otimes F_0((c,d_1)) \longrightarrow F_0((c,d_2)) \,.$$

Definition

(enriched natural transformation)

For $\mathcal{V}$ a cosmos (Def. ), let $\mathcal{C}$ and $\mathcal{D}$ be two $\mathcal{V}$-enriched categories (Def. ) and let

$$\mathcal{C} \underoverset {\underset{G}{\longrightarrow}} {\overset{F}{\longrightarrow}} {} \mathcal{D}$$

be two $\mathcal{V}$-enriched functors (Def. ) from $\mathcal{C}$ to $\mathcal{D}$.

Then a $\mathcal{V}$-enriched natural transformation

$$\mathcal{C} \underoverset {\underset{G}{\longrightarrow}} {\overset{F}{\longrightarrow}} {\phantom{AA}\Downarrow{\mathrlap{\eta}}\phantom{AA}} \mathcal{D}$$

is

• for each $c \in Obj_{\mathcal{C}}$ a choice of morphism

  $$\eta_c \;\colon\; I \longrightarrow \mathcal{D}(F(c),G(c))$$

  such that for each pair of objects $c,d \in \mathcal{C}$ the two morphisms (in $\mathcal{V}$)

(9)$$\eta_d \circ F(-) \;\colon\; \mathcal{C}(c,d) \overset{r}{\simeq} \mathcal{C}(c,d) \otimes I \overset{ G_{c,d} \otimes \eta_c }{\longrightarrow} \mathcal{D}(G(c),G(d)) \otimes \mathcal{D}( F(c), G(c) ) \overset{\circ_{F(c), G(c), G(d)}}{\longrightarrow} \mathcal{D}(F(c), G(d))$$

and

(10)$$G(-) \circ \eta_c \;\colon\; \mathcal{C}(c,d) \overset{\ell}{\simeq} I \otimes \mathcal{C}(c,d) \overset{ \eta_d \otimes F_{c,d} }{\longrightarrow} \mathcal{D}(F(d), G(d)) \otimes \mathcal{D}(F(c), F(d)) \overset{\circ_{F(c),F(d), G(d)}}{\longrightarrow} \mathcal{D}(F(c), G(d))$$

agree.

Example

(functor category of enriched functors)

For $\mathcal{V}$ a cosmos (Def. ) let $\mathcal{C}$, $\mathcal{D}$ be two $\mathcal{V}$-enriched categories (Def. ). Then there is a category (Def. ) of enriched functors (Def. ), to be denoted

$$[\mathcal{C}, \mathcal{D}]$$

whose objects are the enriched functors $\mathcal{C} \overset{F}{\to} \mathcal{D}$ and whose morphisms are the enriched natural transformations between these (Def. ).

In the case that $\mathcal{V} =$ Set, via Def. , with $Set$-enriched categories identified with plain categories via Example , this coincides with the functor category from Example .

Notice that, at this point, $[\mathcal{C}, \mathcal{D}]$ is a plain category, not itself a $\mathcal{V}$-enriched category, unless $\mathcal{V} =$ Set. But it may be enhanced to one, this is Def. below.

There is now the following evident generalization of the concept of adjoint functors (Def. ) from plain category theory to enriched category theory:

Definition

(enriched adjunction)

For $\mathcal{V}$ a cosmos (Def. ), let $\mathcal{C}$, $\mathcal{D}$ be two $\mathcal{V}$-enriched categories (Def. ). Then an adjoint pair of $\mathcal{V}$-enriched functors or enriched adjunction

$$\mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D}$$

is a pair of $\mathcal{V}$-enriched functors (Def. ), as shown, such that there is a $\mathcal{V}$-enriched natural isomorphism (Def. ) between enriched hom-functors (Def. ) of the form

(11)$$\mathcal{C}(L(-),-) \;\simeq\; \mathcal{D}(-,R(-)) \,.$$

Definition

(enriched equivalence of categories)

For $\mathcal{V}$ a cosmos (Def. ), let $\mathcal{C}$, $\mathcal{D}$ be two $\mathcal{V}$-enriched categories (Def. ). Then an equivalence of enriched categories

$$\array{ \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\phantom{AA} \simeq \phantom{AA}} \mathcal{D} }$$

is a pair of $\mathcal{V}$-enriched functors back and forth, as shown (Def. ), together with $\mathcal{V}$-enriched natural isomorphisms (Def. ) between their composition and the identity functors:

$$id_{\mathcal{D}} \overset{\simeq}{\Rightarrow} R \circ L \phantom{AAA} \text{and} \phantom{AAA} L \circ R \overset{\simeq}{\Rightarrow} id_{\mathcal{C}} \,.$$