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\section*{Pr(infinity,1)Cat}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{unstable_version}{Unstable version}\dotfill \pageref*{unstable_version} \linebreak
\noindent\hyperlink{stable_version}{Stable version}\dotfill \pageref*{stable_version} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{adjoint_functor_theorem}{Adjoint functor theorem}\dotfill \pageref*{adjoint_functor_theorem} \linebreak
\noindent\hyperlink{homobjects}{Hom-objects}\dotfill \pageref*{homobjects} \linebreak
\noindent\hyperlink{EmbeddingIntoCat}{Embedding into $(\infty,1)Cat$}\dotfill \pageref*{EmbeddingIntoCat} \linebreak
\noindent\hyperlink{tensor_product}{Tensor product}\dotfill \pageref*{tensor_product} \linebreak
\noindent\hyperlink{colimits}{Colimits}\dotfill \pageref*{colimits} \linebreak
\noindent\hyperlink{as_vector_spaces}{As $\infty$-vector spaces}\dotfill \pageref*{as_vector_spaces} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

$Pr(\infty,1)Cat$ is the [[(∞,1)-category]] of [[locally presentable (∞,1)-categories]] and [[(∞,1)-colimit]]-preserving [[(∞,1)-functors]] between them ([[Lawvere distributions]]).

Recall that a [[presentable (∞,1)-category]] is a [[localization of an (∞,1)-category|localization]] of a [[(∞,1)-category of (∞,1)-presheaves]]. In particular it has all small [[colimits]]. An [[(∞,1)-functor]] $C \times D \to E$ from the cartesian [[product]] of two presentable $(\infty,1)$-categories is \emph{bilinear} if it respects colimits in both variables.

It turns out that there is a \emph{universal} such bilinear functor

\begin{displaymath}
C \times D \to C \otimes D
  \,,
\end{displaymath}
which thereby defines a [[tensor product of presentable (∞,1)-categories]]. This defines a [[monoidal (infinity,1)-category|monoidal structure]] on presentable $(\infty,1)$-categories, which is in fact [[symmetric monoidal (infinity,1)-category|symmetric]].

The collection $Pr(\infty,1)Cat$ of presentable $(\infty,1)$-cateories with colimit-preserving [[(∞,1)-functors]] between them (i.e. with ``\emph{linear}'' functors between them!), is an $(\infty,1)$-generalization of the category $Set Mod$ of ordinary categories and [[bimodules]] or [[profunctors]], or [[distributors]] between them. See [[distributor]] and in particular the discussion there about the equivalent reformulation in terms of colimit-preserving functors.

Using $Pr(\infty,1)Cat$ with its notion of ``linearity'' one obtains a very general notion of $\infty$-linear algebra. This is described at [[geometric ∞-function theory]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{unstable_version}{}\subsubsection*{{Unstable version}}\label{unstable_version}

Write $Pr(\infty,1)Cat_1$ for the [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-categories]] whose

\begin{itemize}%
\item objects are [[presentable (∞,1)-category|presentable (∞,1)-categories]];


\item morphisms are [[(∞,1)-colimit]]-preserving [[(∞,1)-functors]].



\end{itemize}
\hypertarget{stable_version}{}\subsubsection*{{Stable version}}\label{stable_version}

The symmetric monoidal structure on presentable $(\infty,1)$-categories restricts to one on presentable [[stable (∞,1)-category|stable (∞,1)-categories]].

The tensor unit of stable presentable $(\infty,1)$-categories is the [[stable (∞,1)-category of spectra]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{adjoint_functor_theorem}{}\subsubsection*{{Adjoint functor theorem}}\label{adjoint_functor_theorem}

A functor between locally presentable $(\infty,1)$-categories:

\begin{itemize}%
\item is a [[left adjoint]] if and only if it preserves small colimits (hence, in particular, is accessible), and
\item is a [[right adjoint]] if and only if it preserves small limits and is accessible.

\end{itemize}
\hypertarget{homobjects}{}\subsubsection*{{Hom-objects}}\label{homobjects}

\begin{defn}
\label{}\hypertarget{}{}
For $C, D \in Pr(\infty,1)Cat$, let

\begin{displaymath}
Func^L(C,D) \subset Func(C,D)
\end{displaymath}
be the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-functors]] on those that preserve all small [[(∞,1)-colimits]] (equivalently, are left adjoints).

\end{defn}
Evidently $Func^L(C,D)$ would be the hom-$(\infty,1)$-category of an enhancement of $Pr(\infty,1)Cat$ to an $(\infty,2)$-category; its maximal sub-$\infty$-groupoid is the hom-$\infty$-groupoid $Pr(\infty,1)Cat(C,D)$.

\begin{prop}
\label{}\hypertarget{}{}
For all $C,D$ we have that $Func^L(C,D)$ is itself locally presentable.

\end{prop}
This is [[Higher Topos Theory|HTT, prop. 5.5.3.8]].

\hypertarget{EmbeddingIntoCat}{}\subsubsection*{{Embedding into $(\infty,1)Cat$}}\label{EmbeddingIntoCat}

\begin{prop}
\label{LimitsAndColimits}\hypertarget{LimitsAndColimits}{}
All small [[limit in a quasi-category|limits and colimits]] exists in $Pr(\infty,1)Cat$. The limits are preserved by the embedding $Pr(\infty,1)Cat \hookrightarrow$ [[(∞,1)Cat]].

\end{prop}
This is [[Higher Topos Theory|HTT, prop. 5.5.3.13]].

\hypertarget{tensor_product}{}\subsubsection*{{Tensor product}}\label{tensor_product}

\begin{prop}
\label{}\hypertarget{}{}
Let $C_1, \cdots, C_n$ be a finite collection of locally presentable $(\infty,1)$-categories. There exists a locally presentable $(\infty,1)$-category $C_1 \otimes \cdots \otimes C_n$ and an [[(∞,1)-functor]]

\begin{displaymath}
C_1 \times \cdots \times C_n \to C_1 \otimes \cdots \otimes C_n
\end{displaymath}
(the [[tensor product]]) such that

\begin{enumerate}%
\item it preserves [[(∞,1)-colimit]]s in each variable;


\item for every $D \in Pr(\infty,1)Cat$, composition with $f$ produces an [[equivalence of (∞,1)-categories]]



\end{enumerate}
\begin{displaymath}
Func_{(\infty,1)}^L(C_1 \otimes \cdots \otimes C_n,D)
  \stackrel{\simeq}{\to}
  Func^{L}_{(\infty,1)}(C_1 \times \cdots \times C_n,D)
  \hookrightarrow
  Func^{}_{(\infty,1)}(C_1 \times \cdots \times C_n,D)
\end{displaymath}
onto the full [[sub-(∞,1)-category]] of those functors, that preserves colimits in each argument.

\end{prop}
This is (\hyperlink{LurieNoncommutative}{Lurie, NA, theorem 4.1.4}).

This tensor product makes $Pr(\infty,1)Cat$ a [[symmetric monoidal (∞,1)-category]]. Indeed it is even closed, since the hom-objects $Func^L$ supply adjoints to the tensor product.

The definition of the tensor product is easy: $C \otimes D = Cts(C^{op},D)$, where $Cts$ denotes the category of continuous (small-limit-preserving) functors. Equivalently, this is $Cocts(C,D^{op})^{op}$, where $Cocts$ denotes the category of cocontinuous (small-colimit-preserving) functors; it is tempting to write this as $Func^L(C,D^{op})^{op}$, but note that $D^{op}$ is not itself locally presentable. However, $D^{op}$ is nevertheless cocomplete, so any cocontinuous functor $C \to D^{op}$ has a right adjoint.

To show that $Cts(C^{op},D)$ is locally presentable, recall that $C \simeq Ind_\kappa A$ for some small $\kappa$-cocomplete $A$. Thus $Cts(C^{op},D) \simeq \kappa Cts (A^{op},D)$, where $\kappa Cts$ denotes the category of $\kappa$-limit-preserving functors; this is an accessible localization of $Fun(A^{op},D)$ and hence locally presentable.

To show that $Cts(C^{op},D)$ has the right universal property, we compute

\begin{displaymath}
\begin{aligned}
Func^L(C,Func^L(D,E))
&\simeq \kappa CoCts (A,Func^L(D,E)) & \quad \text{(by the universal property of }\; C = Ind_\kappa(A))\\
&\simeq \kappa Cts (A^{op},Func^L(D,E)^{op})^{op} &\quad \text{(by taking opposites)}\\\\
&\simeq \kappa Cts (A^{op},CtsAcc(E,D))^{op} &\quad \text{(by the adjoint functor theorem)}\\
&\simeq CtsAcc(E,\kappa Cts (A^{op},D))^{op} &\quad (*)\\
&\simeq Func^L(\kappa Cts (A^{op},D),E) &\quad \text{(by the adjoint functor theorem)}
\end{aligned}
\end{displaymath}
To justify the step $(*)$, note that both sides are equivalent to some subcategory of functors $A^{op} \times E \to D$, and in both cases part of the requirement is that $\kappa$-limits be preserved in the $A^{op}$-variable and all small limits be preserved in the $E$-variable. The remaining condition on the left is that for each $a\in A$ the corresponding functor $E\to D$ is accessible, i.e. preserves $\lambda_a$-filtered colimits for some cardinal $\lambda_a$. On the right, the remaining condition is that the [[currying|curried]] functor $E\to \kappa Cts (A^{op},D)$ is accessible, i.e. preserves $\mu$-filtered colimits for some cardinal $\mu$. Since $\mu$-filtered colimits commute with $\kappa$-limits as long as $\mu\ge \kappa$, such colimits in $\kappa Cts (A^{op},D)$ are pointwise, so we are at least talking about the same colimits, so the right-hand condition implies the left (take each $\lambda_a = \mu$). On the other hand, because $A$ is small, if the left-hand condition holds we can take $\mu = \sup_{a\in A} \lambda_a$ to make the right-hand condition hold.

\hypertarget{colimits}{}\subsubsection*{{Colimits}}\label{colimits}

Above we remarked that the forgetful functor $Pr(\infty,1)Cat \to (\infty,1)Cat$ preserves limits. It does not preserve colimits, but $Pr(\infty,1)Cat$ does have colimits; they are constructed by passing to right adjoints and taking the limit there.

In particular, this implies there are some cases in which colimits coincide with limits. For instance:

\begin{prop}
\label{}\hypertarget{}{}
Small coproducts in $Pr(\infty,1)Cat$ coincide with small products.

\end{prop}
\begin{proof}
Let $\prod_i C_i$ be a product in $Pr(\infty,1)Cat$. Then we have

\begin{displaymath}
\begin{aligned}
Func^L(\prod_i C_i, D)
&\simeq Cts(D, \prod_i C_i)\\
&\simeq \prod_i Cts(D,C_i)\\
&\simeq \prod_i Func^L(C_i,D)
\end{aligned}
\end{displaymath}
\end{proof}
To be precise, the next result requires an enhancement of $Pr(\infty,1)Cat$ to an $(\infty,2)$-category so that we can talk about [[powers]] and [[copowers]] by small $(\infty,1)$-categories. However, the proof gives an explicit universal property that makes sense without having to define that whole $(\infty,2)$-category.

\begin{prop}
\label{}\hypertarget{}{}
The [[copower]] of $C\in Pr(\infty,1)Cat$ by a small $(\infty,1)$-category $A$ is $C^{A^{op}}$.

\end{prop}
\begin{proof}
Let $P A$ denote the [[presheaf (∞,1)-category]] on $A$. This is its free cocompletion, so we have $Func^L(P A,C) \simeq C^A$. Dually, $(P A)^{op}$ is the free completion of $A^{op}$, so we have $Cts((P A)^{op},C) \simeq C^{A^op}$. Now we have:

\begin{displaymath}
\begin{aligned}
Func^L(C,D)^A
&\simeq Func^L(C,D^A)\\
&\simeq Func^L(C,Func^L(P A,D))\\
&\simeq Func^L(C \otimes P A, D)\\
&\simeq Func^L(Cts((P A)^{op},C),D)\\
&\simeq Func^L(C^{A^op},D)
\end{aligned}
\end{displaymath}
\end{proof}
In general, this sort of argument should work for all lax colimits of lax functors; for instance, [[Kleisli objects]] should also coincide with [[Eilenberg-Moore objects]] (though this also requires enhancing $Pr(\infty,1)Cat$ to an $(\infty,2)$-category). The corresponding 1-categorical fact is that in any [[bicategory]] with [[local colimits]] (colimits in each hom-category distributed over by composition), lax colimits of lax functors are also lax limits.

\hypertarget{as_vector_spaces}{}\subsection*{{As $\infty$-vector spaces}}\label{as_vector_spaces}

In some context it makes good sense to think of $Pr(\infty,1)Cat$ as a model for an $(\infty,1)$-category of ``$\infty$-vector spaces'', or at least ``$\infty$-abelian groups''. For instance, the fact that the $(\infty,2)$-category $Pr(\infty,1)Cat$ has local colimits, making certain limits and colimits coincide, is a sort of categorification of the fact that of [[Vect]] and [[Ab]] are [[additive categories]] in which finite products and coproducts coincide.

More on this analogy is at [[integral transforms on sheaves]]. Here a small $(\infty,1)$-category $S$ is to be thought of as a \emph{basis} and the locally presentable $(\infty,1)$-category $C \hookrightarrow PSh_{(\infty,1)}(C)$ as the $\infty$-vector space spanned by this basis. The colimits in $C$ play the role of addition of vectors and the fact that morphisms in $Pr(\infty,1)Cat$ are colimit-presserving means that they play the role of \emph{linear} maps between vector spaces. This is described also at [[Lawvere distribution]].

The monoidal product $\otimes : Pr(\infty,1)Cat \times Pr(\infty,1)Cat \to Pr(\infty,1)Cat$ plays the role of the [[tensor product]] of vector spaces, with a morphism out of $C \otimes D$ being a bilinear morphism out of $C \times D$, and the fact that $Pr(\infty,1)Cat$ is closed monoidal reflects the fact that [[Vect]] is closed monoidal. The construction of the tensor product as $Cts(C^{op},D)$ corresponds to the fact that, at least for finite-dimensional $V$, we have $V\otimes W \cong Hom(V^\ast,W)$.

(A related decategorification of $Pr(\infty,1)Cat$ is the category [[Sup]] of [[suplattices]], which can also be thought of as analogous to abelian groups or vector spaces.)

\hypertarget{applications}{}\subsubsection*{{Applications}}\label{applications}

Combined with the fact that the embedding $Pr(\infty,1)Cat \hookrightarrow (\infty,1)Cat$ preserves limits (prop. \ref{LimitsAndColimits}), this yields some useful statements.

For instance with $Pr(\infty,1)Cat$ regarded as $\infty Vect$, for any [[∞-group]] $G$ with [[delooping]] [[∞-groupoid]] $\mathbf{B}G$, we may think of an [[(∞,1)-functor]] $\rho : \mathbf{B}G \to Pr(\infty,1)Cat$ as a linear [[representation]] of $G$: the single object of $\mathbf{B}G$ is sent to a presentable $(\infty,1)$-category $V$ and the morphisms in $\mathbf{B}G$ then define an [[action]] of $G$ on that.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[PrCat]]


\item [[Ho(CombModCat)]]


\item [[2Ab]], [[2Ring]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The $(\infty,1)$-category $Pr(\infty,1)Cat$ is introduced in section 5.5.3 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
The monoidal structure on $Pr(\infty,1)Cat$ is described in section 4.1 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Noncommutative Algebra]]}

\end{itemize}
That this is in fact a symmetric monoidal structure is discussed in section 6 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Commutative Algebra]]}

\end{itemize}
(see proposition 6.14 and remark 6.18). This monoidal structure is exhibited as a restriction of a monoidal structure on all cocomplete $(\infty,1)$-categories in section 4.8 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Algebra]]}.

\end{itemize}
[[!redirects symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories]]

[[!redirects symmetric monoidal (∞,1)-category of presentable (∞,1)-categories]] [[!redirects symmetric monoidal (infinity,1)-category of presentable (inf]] [[!redirects Pr(∞,1)Cat]]

category: category



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