2. Preliminaries

Let $\mathcal{V}$ be a symmetric monoidal closed category with the unit object $I$ and the underlying category $\mathcal{V}_0$ . Following Kelly (Reference Kelly1982), $\mathcal{V}$ is locally $\lambda$ -presentable as a symmetric monoidal closed category if $\mathcal{V}_0$ is locally $\lambda$ -presentable, $I$ is $\lambda$ -presentable in $\mathcal{V}_0$ , and $X\otimes Y$ is $\lambda$ -presentable in $\mathcal{V}_0$ whenever $X$ and $Y$ are $\lambda$ -presentable in $\mathcal{V}_0$ . This ensures that $\lambda$ -presentable objects in the enriched sense and in the ordinary sense coincide. We will denote by $\mathcal{V}_\lambda$ the (representative) small, full subcategory consisting of $\lambda$ -presentable objects.

The underlying functor $\mathcal{V}_0(I,-):\mathcal{V}_0\to \textbf {Set}$ has a left adjoint $-\cdot I$ sending a set $X$ to the coproduct $X\cdot I$ of $X$ copies of $I$ in $\mathcal{V}_0$ . Objects $X\cdot I$ will be called discrete (see Rosický (Reference Rosický2024)). Every object $V$ of $\mathcal{V}$ determines a discrete object $V_0=\mathcal{V}_0(I,V)\cdot I$ and morphisms $\delta _V:V_0\to V$ given by the counit of the adjunction. Every morphism $f:V\to W$ determines the morphism $f_0=\mathcal{V}_0(I,f)\cdot I$ between the underlying discrete objects. We will denote by $\mathcal{D}_\lambda$ the (representative) full subcategory consisting of discrete $\lambda$ -presentable objects. If $\mathcal{V}_0(I,-)$ preserves $\lambda$ -presentable objects, then $|\mathcal{V}_0(I,X)|\lt \lambda$ for every $\lambda$ -presentable object $X$ . Hence $X_0$ is $\lambda$ -presentable as well.

A morphism $f:A\to B$ will be called a surjection if $\mathcal{V}_0(I,f)$ is surjective (see Rosický (Reference Rosický2024)). For instance, every $\delta _V$ is surjective. Let $Surj$ denote the class of all surjections in $\mathcal{V}_0$ and let $Inj$ be the class of morphisms of $\mathcal{V}_0$ having the unique right lifting property w.r.t. every surjection. Morphisms from $Inj$ will be called injections. $(Surj,Inj)$ is a factorization system in $\mathcal{V}_0$ if and only if $Surj$ is closed under colimits (see Rosický (Reference Rosický2024, 3.3)). It is a $\mathcal{V}$ -factorization system if and only if $Surj$ is moreover closed under tensors (Lucyshyn-Wright, Reference Lucyshyn-Wright2014, 5.7). Surjections are epimorphisms provided that $I$ is a generator because then $\mathcal{V}(I,-)$ is faithful, thus reflecting epimorphisms.

Let $\mathcal{A}$ be a small, full, dense sub- $\mathcal{V}$ -category of $\mathcal{V}$ with the inclusion $K:\mathcal{A}\to \mathcal{V}$ . Objects of $\mathcal{A}$ are called arities. Then an $\mathcal{A}$ -pretheory is an identity-on-objects $\mathcal{V}$ -functor $J:\mathcal{A}\to \mathcal{T}$ . A $\mathcal{T}$ -algebra is an object $A$ of $\mathcal{V}$ together with a $\mathcal{V}$ -functor $\hat {A}:\mathcal{T}^{\mathrm{op}}\to \mathcal{V}$ whose composition with $J^{\mathrm{op}}$ is $\mathcal{V}(K-,A)$ . Hence every morphism $f:JY\to JX$ induces an $(X,Y)$ -ary operation $\tilde {A}(f):A^X\to A^Y$ on $A$ . Every $\mathcal{A}$ -pretheory $\mathcal{T}$ induces a $\mathcal{V}$ -monad $T:\mathcal{V}\to \mathcal{V}$ given by its $\mathcal{V}$ -category $\textbf {Alg}(\mathcal{T})$ of algebras. Conversely, a $\mathcal{V}$ -monad $T$ induces an $\mathcal{A}$ -pretheory $J:\mathcal{A}\to \mathcal{T}$ , where $\mathcal{T}$ is the full subcategory of $\textbf {Alg}(T)$ consisting of free algebras on objects from $\mathcal{A}$ and $J$ is the domain-codomain restriction of the free algebra functor. An $\mathcal{A}$ -pretheory is an $\mathcal{A}$ -theory if it is given by its monad. Then $\hat {A}$ is the hom-functor $\textbf {Alg}(\mathcal{T})(-,A)$ restricted to free algebras over $\mathcal{A}$ . On the other hand, a monad is called $\mathcal{A}$ -nervous if it is given by its theory. All this is explained in Bourke and Garner (Reference Bourke and Garner2019) where $\mathcal{A}$ -theories and $\mathcal{A}$ -nervous monads are characterized ( $\mathcal{T}$ -algebras are called concrete $\mathcal{T}$ -models).

Under a $\lambda$ -ary $\mathcal{V}$ -theory, we will mean a $\mathcal{V}_\lambda$ -theory. Following Bourke and Garner (Reference Bourke and Garner2019), $\lambda$ -ary $\mathcal{V}$ -theories correspond to $\mathcal{V}$ -monads on $\mathcal{V}$ preserving $\lambda$ -filtered colimits. They are called $\lambda$ -ary $\mathcal{V}$ -monads. Since $\mathcal{D}_\lambda$ is dense in $\mathcal{V}$ , we can consider $\mathcal{D}_\lambda$ -theories that correspond to $\lambda$ -ary discrete Lawvere theories of Power (Reference Power2005). On $\textbf {Met}$ , finitary discrete Lawvere theories correspond to varieties of unconditional quantitative theories of Mardare et al. (Reference Mardare, Panangaden and Plotkin2016, Reference Mardare, Panangaden and Plotkin2017) (see Rosický (Reference Rosický2021, 4.7)). On the category $\textbf {Pos}$ of posets, finitary discrete Lawvere theories correspond to varieties of ordered algebras of Bloom (Reference Bloom1976) (see Rosický (Reference Rosický2021, 4.7)).

We will say that a $\mathcal{V}$ -functor $H:\mathcal{V}\to \mathcal{V}$ is strongly $\lambda$ -ary if it is $\mathrm{Lan}_K HK$ , where $K:\mathcal{D}_\lambda \to \mathcal{V}$ is the inclusion (see Kelly and Lack (Reference Kelly and Lack1993) for $\lambda =\aleph _0)$ . A $\mathcal{V}$ -monad $T$ is strongly $\mu$ -ary if $T$ is strongly $\mu$ -ary as a $\mathcal{V}$ -functor. Let $\tilde {K}:\mathcal{V}\to [(\mathcal{D}_\lambda )^{\mathrm{op}},\mathcal{V}]$ be the induced fully faithful $\mathcal{V}$ -functor.

3. Strongly $\lambda$ -Ary Monads

Recall that an object $A$ in a category $\mathcal{K}$ is $\mu$ -generated if its hom-functor $\mathcal{K}(A,-):\mathcal{K}\to \textbf {Set}$ preserves $\mu$ -directed colimits of monomorphisms. A cocomplete category $\mathcal{K}$ is locally $\mu$ -generated if it has a set $\mathcal{A}$ of $\mu$ -generated objects such that every object is a $\mu$ -directed colimit of its subobjects from $\mathcal{A}$ (see Adámek and Rosický (Reference Adámek and Rosický1994)). As in the locally presentable case, $\mathcal{V}$ is locally $\mu$ -generated as a symmetric monoidal closed category if $\mathcal{V}_0$ is locally $\mu$ -generated, $I$ is $\mu$ -generated in $\mathcal{V}_0$ , and $X\otimes Y$ is $\mu$ -generated in $\mathcal{V}_0$ whenever $X$ and $Y$ are $\mu$ -generated in $\mathcal{V}_0$ . This ensures that $\mu$ -generated objects in the enriched sense and in the ordinary sense coincide.

Proof. Let $T$ be a strongly $\mu$ -ary $\mathcal{V}$ -monad on $\mathcal{V}$ . Since $\mathcal{D}_\mu$ consists of $\lambda$ -presentable objects, $T$ preserves $\lambda$ -directed colimits (see Kelly (Reference Kelly1982, 5.29)). Following Bourke and Garner (Reference Bourke and Garner2019), $\mathcal{V}^T$ is equivalent to $\textbf {Alg}(\mathcal{T})$ where $\mathcal{T}$ is an $\lambda$ -ary $\mathcal{V}$ -theory whose $(X,Y)$ -ary operations correspond to morphisms $f:FY\to FX$ ; here, $U:\textbf {Alg}(T)\to \mathcal{V}$ is the forgetful functor, and $F$ is its enriched left adjoint.

We will show that $T$ preserves surjections. Let $f:X\to Y$ be a surjection. Express $Y$ and $X$ as weighted colimits $\tilde {K}X\ast K$ and $\tilde {K}Y\ast K$ of $\mu$ -presentable discrete objects. Following Kelly (Reference Kelly1982, 5.29), $T$ preserves these weighted colimits, that is, $TX=\tilde {K}X\ast TK$ and $TY=\tilde {K}Y\ast TK$ . Hence

\begin{equation*} T(f)=\tilde {K}(f)\ast TK. \end{equation*}

Since

\begin{equation*} \tilde {K}(f)_{Z\cdot I}= [Z\cdot I,f]= f^Z \end{equation*}

and surjections are closed under products (Rosický Reference Rosický2024, 3.2(2)), $\tilde {K}(f):\tilde {K}X\to \tilde {K}Y$ is pointwise surjective. The dual of Lucyshyn-Wright (Reference Lucyshyn-Wright2014, 4.5) implies that $T(f)$ is surjective.

Let $\mathcal{T}_d$ be the restriction of $\mathcal{T}$ on $\mathcal{D}_\lambda$ and

\begin{equation*} R: \textbf {Alg}(\mathcal{T})\to \textbf {Alg}(\mathcal{T}_d) \end{equation*}

the reduct functor. Clearly, it is a $\mathcal{V}$ -functor, and we will prove that it is an isomorphism. At first, we will show that $R$ is an embedding. It is clearly faithful. Following Rosický (Reference Rosický2024, 4.4), for every $f:FY\to FX$ , there is $f_0:FY_0\to FX_0$ such that $F(\delta _X)f_0=fF(\delta _Y)$ . Let $\omega$ be an $(X,Y)$ -ary operation corresponding to $f:FY\to FX$ and $\omega _0$ an $(X_0,Y_0)$ -ary operation corresponding to $f_0$ . Consider a $\mathcal{T}$ -algebra $A$ . We have

\begin{equation*} (UA)^{\delta _Y}\omega _A=(\omega _0)_A(UA)^{\delta _X} \end{equation*}

Since $I$ is a generator, $\delta _Y$ is an epimorphism, and thus, $(UA)^{\delta _Y}$ is a monomorphism. Hence $(\omega _0)_A=(\omega _0)_B$ implies that $\omega _A=\omega _B$ for any $\mathcal{T}$ -algebras $A$ and $B$ . Thus, $R$ is an embedding.

We will show that $R$ is surjective on objects. Let $A$ be a $\mathcal{T}_d$ -algebra. Again, let $\omega$ be an $(X,Y)$ -ary operation corresponding to $f:FY\to FX$ and $\omega _0$ an $(X_0,Y_0)$ -ary operation corresponding to $f_0$ . Since $\mathcal{V}_0(I,-)$ preserves $\lambda$ -presentable objects, $X_0$ and $Y_0$ are $\lambda$ -presentable. Since $A$ is an $\mathcal{T}_d$ -algebra, we get

\begin{equation*} (\omega _0)_A=\tilde {A}(f_0):(UA)^{X_0}\to (UA)^{Y_0}. \end{equation*}

We will show that there is a unique $\,\tilde{\!f}:(UA)^{X_0}\to (UA)^Y$ such that $(UA)^{\delta _Y}\,\tilde{\!f}=(\omega _0)_A$ . Consider $\varphi :\tilde {K}Y\to [(UA)^{X_0},(UA)^K]$ sending $a:Z\cdot I\to Y$ to

\begin{equation*} (UA)^{X_0} \xrightarrow {\ (\omega _0)_A } (UA)^{Y_0} \xrightarrow {\ (UA)^{a_0}} (UA)^{Z\cdot I} \end{equation*}

where $a_0:Z\cdot I\to Y_0$ is determined by $a$ , that is, $\delta _Y a_0=a$ . Since $(UA)^Y$ is a weighted limit $\{\tilde {K}Y,(UA)^K\}$ , we get $\,\tilde{\!f}:(UA)^{X_0}\to (UA)^Y$ induced by $\varphi$ . Then $(UA)^{\delta _Y}\,\tilde{\!f}=\{\tilde {K}\delta _Y,(UA)^K\}\,\tilde{\!f}$ is induced by $\varphi \tilde {K}\delta _Y$ . Put $\omega _A=\,\tilde{\!f}(UA)^{\delta _X}$ . We have

\begin{equation*} (UA)^{\delta _Y}\omega _A=(UA)^{\delta _Y}\,\tilde{\!f}(UA)^{\delta _X}=(\omega _0)_A(UA)^{\delta _X}. \end{equation*}

We have to show that this makes $A$ a $\mathcal{T}$ -algebra. Let $\rho$ be a $(Y,Z)$ -ary operation corresponding to $g:FY\to FY$ . We have to show that $(\rho \omega )_A=\rho _A\omega _A$ . Put $h=gf$ . Then

\begin{equation*} \rho _A\omega _A=\tilde {g}(UA)^{\delta _Y}\,\tilde{\!f}(UA)^{\delta _X}= \tilde {g}(\omega _0)_A(UA)^{\delta _X}. \end{equation*}

Since

\begin{equation*} (UA)^{\delta _Z}\tilde {g}(\omega _0)_A=(\rho _0)_A(\omega _0)_A=((\rho \omega )_0)_A=(UA)^{\delta _Z}\tilde {h}, \end{equation*}

the unicity implies that $\tilde {h}=\tilde {g}(\omega _0)_A$ . Hence

\begin{equation*} (\rho \omega )_A=\tilde {h}(UA)^{\delta _X}=\tilde {g}(\omega _0)_A(UA)^{\delta _X}=\tilde {g}(UA)^{\delta _Y}\omega _A=\rho _A\omega _A. \end{equation*}

This proves that the reduct functor $R$ is surjective on objects. It remains to show that $R$ is full. Consider $\mathcal{T}$ -algebras $A$ and $B$ and a homomorphism $h:RA\to RB$ . Then

\begin{equation*} (UB)^{\delta _Y}h^Y\omega _A=h^{Y_0}(UA)^{\delta _Y}\omega _A=h^{Y_0}(\omega _0)_A(UA)^{\delta _X}=(\omega _0)_Bh^{X_0}(UA)^{\delta _X} \end{equation*}

and

\begin{equation*} (UB)^{\delta _Y}\omega _Bh^X=(\omega _0)_B(UB)^{\delta _X}h^X=(\omega _0)_Bh^{X_0}(UA)^{\delta _X} \end{equation*}

Since $(UB)^{\delta _Y}$ is a monomorphism, we get

\begin{equation*} h^Y\omega _A=\omega _Bh^X. \end{equation*}

Thus, $h:A\to B$ is a homomorphism. We have proved that $R$ is full.

Since $\mathcal{D}_\mu$ consists of $\mu$ -generated objects, $T$ preserves $\mu$ -directed colimits of monomorphisms (more precisely, it sends $\mu$ -directed colimits of monomorphisms to $\mu$ -directed colimits, see Kelly (Reference Kelly1982, 5.29)). Let $\mathcal{T}'_d$ be a subtheory of $\mathcal{T}_d$ consisting of $(X,Y)$ -ary operations, where $X$ is discrete $\lambda$ -presentable and $Y$ discrete $\mu$ -presentable. Rosický (Reference Rosický2021, 3.17) implies that the reduct $\mathcal{V}$ -functor

\begin{equation*} \textbf {Alg}(\mathcal{T}_d)\to \textbf {Alg}(\mathcal{T}'_d) \end{equation*}

is an equivalence.

Let $\mathcal{T}_{d\mu }$ be the restriction of $\mathcal{T}_d$ on $\mathcal{D}_\mu$ . Every discrete $\lambda$ -presentable object $X$ is a $\mu$ -directed colimit $x_i:X_i\to X$ of discrete $\mu$ -presentable objects $X_i$ and split monomorphisms. Consider an $(X,Y)$ -ary operation of $\mathcal{T}'_d$ corresponding to a morphism $f:FY\to FX$ , hence to its adjoint transpose $f^\ast :Y\to TX$ . Since $Tx_i:TX_i\to TX$ is a $\mu$ -directed colimit of $TX_i$ and split monomorphisms, $f^\ast$ factorizes through some $Tx_i$ as $f^\ast =(Tx_i)g$ . Then $g:Y\to TX_i$ yields an $(X_i,Y)$ -ary operation $g^\ast$ . Consequently, the reduct $\mathcal{V}$ -functor

\begin{equation*} \mathcal{T}'_d\to \mathcal{T}_{d\mu } \end{equation*}

is an equivalence. Hence the reduct functor $\textbf {Alg}(\mathcal{T})\to \textbf {Alg}(\mathcal{T}_{d\mu })$ is a $\mathcal{V}$ -equivalence. Thus, $T$ is given by a $\mu$ -ary discrete Lawvere theory $\mathcal{T}_{d\mu }$ .

4. Discrete Lawvere Theories

Following Bourke and Garner (Reference Bourke and Garner2019), a signature is a functor $ \mathrm{ob}\ \mathcal{D}_\mu \to \mathcal{V}$ where $\mathrm{ob}\mathcal{D}_\mu$ is the discrete $\mathcal{V}$ -category on the set of objects of $\mathcal{D}_\mu$ . Thus, a signature $\Sigma$ specifies for each discrete $\mu$ -presentable object $X\cdot I$ an object $\Sigma X$ of operations of input arity $X$ , that is, of $(X,I)$ -ary operations. A signature $\Sigma$ will be called discrete if all objects $\Sigma X$ are discrete.

Proof. Following the proof of Bourke and Garner (Reference Bourke and Garner2019, Proposition 55), a free $\mathcal{V}$ -monad $T_\Sigma$ over $\Sigma$ is a free monad over the functor

\begin{equation*} \coprod \limits _X (-^X)\cdot \Sigma X \end{equation*}

where $|X|\lt \mu$ . Since $\Sigma X$ is discrete, $T_\Sigma$ is a coproduct of powers $-^X$ , $|X|\lt \mu$ . Since strongly $\mu$ -ary functors are closed under coproducts (Bourke and Garner Reference Bourke and Garner2019, Lemma 57), it suffices to show that powers $-^X$ , $|X|\lt \mu$ are strongly $\mu$ -ary. But this follows from the enriched Yoneda lemma because, for a $\mathcal{V}$ -functor $S:\mathcal{V}\to \mathcal{V}$ , we have

\begin{equation*} [(\mathcal{D}_\mu )^{\mathrm{op}},\mathcal{V}](\tilde {K}X,SK)\cong SX\cong [\mathcal{V},\mathcal{V}](\mathcal{V}(KX,-),S). \end{equation*}