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Today I encountered the notion of multicolimit.

Lacking a standard reference for this notion, let me give a self-contained definition of this gadget.

If $S\colon \cal K\to E$ is a diagram, we define its multicolimit as a small set of cocones $\{S\xrightarrow{\varphi_i} L_{i,S}\}_{i\in I}$ such that for any other cone $S \xrightarrow{\delta} \Delta_X $ in $\cal E$ ($\Delta_X$ the constant functor on $X$) there exists a unique index $i(\delta)\in I$ such that $\delta$ factors uniquely through $\varphi_{i(\delta)}$. This explicit definition can be summarized asking that the functor $\cal E\to Set$ which sends any object $E\in\cal E$ to the set of cocones for $S$ with summit $E$ is isomorphic to a small coproduct of representables: $$ [{\cal K,E}](S,\Delta_E)\cong \coprod_{i\in I}{\cal E}(L_{i,S},E) $$ I am interested in understanding if this notion in a genuine generalization of the notion of colimit for $S$, and in establishing some formal properties for such an object. In particular:

  1. Can the notion of multicolimit for $S$ be reduced to a suitable (weighted? weak?) colimit? My sensation is that this notion is utterly different, but ...
  2. ...I'm wondering if the notation $\{\varinjlim\!{}^i S\}_i$, chosen as a pure portmanteau, meaningful to denote the multicolimit of $S$ (provided that we are aware that each $\varinjlim\!{}^i S$ is a colimit only on a suitable (possibly empty) restriction of $S$)?
  3. Assume that $S\colon \cal C\times D\to E$ is a functor, and that each $\varinjlim\!{}^i_{\cal D} S(c,-)$, $\varinjlim\!{}^j_{\cal C} S(-,d)$, $\varinjlim\!{}^{(i,j)}_{\cal C\times D} S$ exists. Do multicolimits commute with multicolimits? In other words, is it true that $$ \varinjlim\!{}^i_{\cal D}\varinjlim\!{}^j_{\cal C} S(c,d)\cong \varinjlim\!{}^j_{\cal C}\varinjlim\!{}^i_{\cal D} S(c,d)\cong \varinjlim\!{}^{(i,j)}_{\cal C\times D} S(c,d) $$
  4. Do left adjoints preserve multicolimits?
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For the moment the only statement which seems true to me is the fourth. – tetrapharmakon Dec 24 '13 at 21:27
up vote 6 down vote accepted
  1. Yes, but perhaps in an other sense than you may think.
  2. I have nothing to say :-)
  3. Yes.
  4. Yes. $\newcommand{\mor}[3]{#1 \colon #2 \rightarrow #3}% \newcommand{\catl}[1]{\mathbb{#1}}% \newcommand{\catw}[1] {\mathbf{#1}}$

Here is my elaboration.

Let me first omit some irrelevant details. We shall say that a functor $\mor{F}{\catl{C}}{\catl{D}}$ has a left multiadjoint if for every $X \in \catl{D}$, the hom-functor $\hom(X, F(-))$ is a (small) coproduct of representables: $$\hom(X, F(-)) \approx \coprod_i\hom(G_i(X), -)$$ At this point I am not sure yet if I fully understand the above definition --- it makes me wonder if there is anything so special about coproducts: i.e. what if we substituted copoducts with other classes of colimits? Let me try:

  • A functor $\mor{F}{\catl{C}}{\catl{D}}$ has a left adjoint if for every $X \in \catl{D}$, the hom-functor $\hom(X, F(-))$ is representable: $$\hom(X, F(-)) \approx \hom(G(X), -)$$ I have nothing to add here.

  • A functor $\mor{F}{\catl{C}}{\catl{D}}$ has a left nothingadjoint if for every $X \in \catl{D}$ the hom-functor $\hom(X, F(-))$, is a (small) colimit of representables: $$\hom(X, F(-)) \approx \mathit{colim}_i\hom(G_i(X), -)$$ Nothingadjointness is not an interesting concept, because by Yoneda every $\catw{Set}$-valued functor is a colimit of representables.

  • A functor $\mor{F}{\catl{C}}{\catl{D}}$ has a finitary approximation to a left adjoint if for every $X \in \catl{D}$, the hom-functor $\hom(X, F(-))$ is a filtered colimit of representables: $$\hom(X, F(-)) \approx \mathit{fcolim}_i\hom(G_i(X), -)$$ In the literature, a functor that has a finitary approximation to a left adjoint is called (left) flat.

Another way of looking at the above three situations is that we want to "represent" objects (apologize for contravariance, but I have not realized that I am describing a contravariant world until now) from $\catw{Set}^\catl{C}$ in $\catl{C}^{op}$, in $\catw{Set}^\catl{C}$ (i.e. a free cocompletion of $\catl{C}^{op}$) and in $\mathit{Ind}(\catl{C}^{op})$ (i.e. a free cofiltered completion of $\catl{C}^{op}$) respectively. A reasonable person, who wants to better understand multiadjunctions, should start looking now for free coproduct completion of a category.

Let $\catl{B}$ be a locally small category. One may associate with it the canonical $\catw{Set}$-indexing functor $\mor{\mathit{fam}(\catl{B})}{\catw{Set}^{op}}{\catw{Cat}}$: $$X \mapsto \catl{B}^X$$ One may think of $\catl{B}^X$ as of the category of formal $X$-indexed coproducts of objects from $\catl{B}$. The Grothendieck construction for $\mathit{fam}(\catl{B})$ glues categories $\catl{B}^X$ of formal $X$-indexed coproducts along sets $X$, giving the category: $$\int \mathit{fam}(\catl{B})$$ which is a formal (small) coproduct completion of $\catl{B}$ (of course, one needs to carefully check this statement). In fact, the above construction rises to a monad on $\catw{Cat}$, and one may develop a formal theory of multiadjunctions (similar to the formal theory of adjunctions through distributors) in the 2-category of Kleisly resolution of the monad.

Nonetheless, there is a less heavy explanation. I claim that a functor $\mor{F}{\catl{C}}{\catl{D}}$ has a left multiadjoint if $\mor{{F^{op}}^\star}{\int \mathit{fam}(\catl{C}^{op})}{\int \mathit{fam}(\catl{D}^{op})}$ defined as: $${F^{op}}^\star(\{C_i\}_{i \in I}) = \{F^{op}(C_i)\}_{i \in I}$$ has right adjoint. First, let me show the trivial direction --- assume that ${F^{op}}^\star$ has right adjoint. In particular, this gives us: $$\hom_{\int \mathit{fam}(\catl{D}^{op})}({F^{op}}^\star(\{C_i\}_{i \in 1}), \{D_j\}_{j \in 1}) \approx \hom_{\int \mathit{fam}(\catl{C}^{op})}(\{C_i\}_{i \in 1}, G(\{D_j\}_{j \in 1}))$$ which simplifies to: $$\hom_{\int \mathit{fam}(\catl{D}^{op})}(F^{op}(C), \{D\}) \approx \hom_{\int \mathit{fam}(\catl{C}^{op})}(\{C\}, G(\{D\}))$$ A morphism $F^{op}(C) \rightarrow \{D\}$ in $\int \mathit{fam}(\catl{D}^{op})$ is just a morphism $D \rightarrow F(C)$ in $\catl{D}$. Similarly, a single morphism $\{C\} \rightarrow G(\{D\})$ in $\int \mathit{fam}(\catl{C}^{op})$ is a morphism $G(\{D\})_k \rightarrow C$ in $\catl{C}$, for one $k \in K$, where $K$ is the indexing set of $G(\{D\})$. So: $$\hom_{\int \mathit{fam}(\catl{C}^{op})}(\{C\}, G(\{D\}))\approx \coprod_{k\in K} \hom_\catl{C}(G(\{D\})_k, C)$$ and we get the formula for left multiadjunction: $$\hom_\catl{D}(F(C), D) \approx \coprod_{k\in K} \hom_\catl{C}(G(\{D\})_k, C)$$ In the other direction, let us assume that the above formula holds, and freely extend $G$ to ${G^{op}}^\star$: $${G^{op}}^\star(\{D_j\}_{j \in J}) = \coprod_{j\in J} G(\{D_j\})$$ Since ${F^{op}}^\star$ is free, it suffices to show: $$\hom_{\int \mathit{fam}(\catl{D}^{op})}({F^{op}}^\star(\{C\}), \{D_j\}_{j \in J}) \approx \hom_{\int \mathit{fam}(\catl{C}^{op})}(\{C\}, {G^{op}}^\star(\{D_j\}_{j \in J}))$$ The left side is isomorphic to: $$\coprod_{j \in J} \hom_\catl{D}(D_j, F(C))$$ whereas, the right side is isomorphic to: $$\coprod_{j \in J} \coprod_{k \in K} \hom_\catl{C}(G(\{D_j\})_k, C)$$ Thus ${G^{op}}^\star$ is right adjoint to ${F^{op}}^\star$.

Moving back to your questions:

  1. Yes, they are colimits in the free coproduct completion of a category.
  2. Still nothing to say :-)
  3. Yes, because it is generally true for (co)limits.
  4. Yes, because the coproduct completion, being a 2-functor, preserves adjunctions.
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Michal, let me say that your answers are always enlightening. I was near to rediscover the coproduct-completion characterization, but I was miles away from this incredible clarity. Thank you! – tetrapharmakon Dec 25 '13 at 0:58
Thanks, I'm happy that I'm useful :-) – Michal R. Przybylek Dec 25 '13 at 9:53
@tetrapharmakon, I thought that you invented the notion of a multicolimit in the above question. Now, I've found that this notion is due to Y. Diers ("Categories localement multipresentables", Arch. Math. 34 (1980)). Silly me! – Michal R. Przybylek Dec 25 '13 at 19:04
In fact I took it from the famous paper by Adamek, Rosicki, Lack, Borceux… about D-presentable categories. They claim that each component of a multicolimit of D-presentable objects is itself D-presentable, and this captured my attention. Since I'm a trying to variate on this theme, I'm particularly interested in your generalization to different classes of colimits! – tetrapharmakon Dec 25 '13 at 20:06
Maybe it's a bit unpolite, but I'll try to ask you: can you elaborate a bit on the construction of $\int fam(\mathbb B)$ as a monad on $\bf Cat$? Would you mind if we talk privately about this (via email, or any chat you want)? Just to give less inertia to the discussion :) – tetrapharmakon Dec 25 '13 at 20:10

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