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?
  • $\begingroup$ For the moment the only statement which seems true to me is the fourth. $\endgroup$
    – fosco
    Dec 24, 2013 at 21:27

1 Answer 1

  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.
  • 1
    $\begingroup$ 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! $\endgroup$
    – fosco
    Dec 25, 2013 at 0:58
  • $\begingroup$ @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! $\endgroup$ Dec 25, 2013 at 19:04
  • $\begingroup$ In fact I took it from the famous paper by Adamek, Rosicki, Lack, Borceux citeseerx.ist.psu.edu/viewdoc/… 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! $\endgroup$
    – fosco
    Dec 25, 2013 at 20:06
  • $\begingroup$ 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 :) $\endgroup$
    – fosco
    Dec 25, 2013 at 20:10
  • $\begingroup$ @tetrapharmakon, do not hesitate (but expect latency). I always write to people if I want anything from them, so, I guess, it's my duty to respond on similar requests :-) You'll find my e-mail on my web-site. $\endgroup$ Dec 25, 2013 at 21:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.