Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorTom Hirschowitz
   • CommentTimeNov 25th 2011
   • (edited Nov 25th 2011)
   • PermaLink
   Author: Tom Hirschowitz
   Format: MarkdownItexHi all, I'm meeting a weird universal property in CAT, and started wondering whether the enriched toolbox would apply. I have functors $L \colon A \to B$, $R \colon A \to C$, $F \colon B \to sets$, $G \colon C \to sets$, and $H \colon A \to sets$, plus natural transformations $\alpha \colon FL \to H$ and $\beta \colon GR \to H$. I have been considering the following two universal properties. 1. The pullback in the (op?)lax slice CAT / sets (I hope you can guess what I mean by lax slice). 2. Same as (1), but imposing furthermore that the domain be 1, i.e., that the universal object is a set. I think there are various ways to calculate these, the most obvious, for (2), being to calculate the limits of all functors to reflect the whole diagram in sets (actually [1,sets]), and then compute the limit there. Another being to calculate (1), and then take the limit (more precisely the right extension along the unique functor to 1). I'm interested in whether these universal properties are definable by weighted colimits in CAT, or as limits in sets seen as a complete object of the 2-category CAT with its standard Yoneda structure. Notably, are there results explaining why the various ways of computing these limits coincide?

   Hi all, I’m meeting a weird universal property in CAT, and started wondering whether the enriched toolbox would apply.

   I have functors

   $L \colon A \to B$, $R \colon A \to C$, $F \colon B \to sets$, $G \colon C \to sets$, and $H \colon A \to sets$,

   plus natural transformations $\alpha \colon FL \to H$ and $\beta \colon GR \to H$.

   I have been considering the following two universal properties.

   1. The pullback in the (op?)lax slice CAT / sets (I hope you can guess what I mean by lax slice).

   2. Same as (1), but imposing furthermore that the domain be 1, i.e., that the universal object is a set.

   I think there are various ways to calculate these, the most obvious, for (2), being to calculate the limits of all functors to reflect the whole diagram in sets (actually [1,sets]), and then compute the limit there. Another being to calculate (1), and then take the limit (more precisely the right extension along the unique functor to 1).

   I’m interested in whether these universal properties are definable by weighted colimits in CAT, or as limits in sets seen as a complete object of the 2-category CAT with its standard Yoneda structure. Notably, are there results explaining why the various ways of computing these limits coincide?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeNov 25th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThe pullback of what two morphisms in $CAT/set$? I see a span in $CAT/set$ ,not a cospan; was that a typo?

   The pullback of what two morphisms in $CAT/set$? I see a span in $CAT/set$ ,not a cospan; was that a typo?

3. • CommentRowNumber3.
   • CommentAuthorTom Hirschowitz
   • CommentTimeNov 25th 2011
   • PermaLink
   Author: Tom Hirschowitz
   Format: TextYou're probably right that morphisms should be oriented according to the 1-cell, not the 2-cell. So yes, it's a typo, and pushout, etc.

   You're probably right that morphisms should be oriented according to the 1-cell, not the 2-cell. So yes, it's a typo, and pushout, etc.
4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeNov 26th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYou mean, what you want is a pushout in the lax slice 2-category? I also don't quite understand what you mean by "imposing that the domain be 1". Maybe you could give an example?

   You mean, what you want is a pushout in the lax slice 2-category? I also don’t quite understand what you mean by “imposing that the domain be 1”. Maybe you could give an example?

5. • CommentRowNumber5.
   • CommentAuthorTom Hirschowitz
   • CommentTimeNov 28th 2011
   • PermaLink
   Author: Tom Hirschowitz
   Format: MarkdownItexSorry, I don't seem to explain myself well. Yes, what I want is a pushout in the lax slice 2-category (although I'm not sure the '2' matters here). By "imposing that the domain be 1", I mean the same as when computing a Kan extension along the unique 1-cell to 1. That gives perhaps the best possible example: are, e.g., right extensions of a 1-cell, say $f \colon C \to D$, along another particular examples of limits in the ambient 2-category (as opposed to in $D$)? The original example was similar, but with interchange of limits: given functors $F \colon C \to sets$, $G \colon D \to sets$, and $H \colon C \to D$ such that $GH = F$, one would compute limF, limG, and the induced map $limH \colon limG \to limF$, and then pull limH back along some element $x \colon 1 \to limF$ to obtain a set $X$. This element induces a transformation $\alpha \colon x! \to F$, and I think $X$ may be obtained alternatively by first computing the pushout of $\alpha$ and $H$ in the lax slice $CAT / sets$, and then taking the limit of the resulting functor. Is this clearer?

   Sorry, I don’t seem to explain myself well.

   Yes, what I want is a pushout in the lax slice 2-category (although I’m not sure the ’2’ matters here).

   By “imposing that the domain be 1”, I mean the same as when computing a Kan extension along the unique 1-cell to 1. That gives perhaps the best possible example: are, e.g., right extensions of a 1-cell, say $f \colon C \to D$, along another particular examples of limits in the ambient 2-category (as opposed to in $D$)?

   The original example was similar, but with interchange of limits: given functors $F \colon C \to sets$, $G \colon D \to sets$, and $H \colon C \to D$ such that $GH = F$, one would compute limF, limG, and the induced map $limH \colon limG \to limF$, and then pull limH back along some element $x \colon 1 \to limF$ to obtain a set $X$. This element induces a transformation $\alpha \colon x! \to F$, and I think $X$ may be obtained alternatively by first computing the pushout of $\alpha$ and $H$ in the lax slice $CAT / sets$, and then taking the limit of the resulting functor.

   Is this clearer?

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeNov 28th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex> By "imposing that the domain be 1", I mean the same as when computing a Kan extension along the unique 1-cell to 1. What I was looking for was a precise statement of the universal property you are after. Your mention of Kan extensions gives me some vague idea of what you might mean, but not a precise one.

   By “imposing that the domain be 1”, I mean the same as when computing a Kan extension along the unique 1-cell to 1.

   What I was looking for was a precise statement of the universal property you are after. Your mention of Kan extensions gives me some vague idea of what you might mean, but not a precise one.

7. • CommentRowNumber7.
   • CommentAuthorTom Hirschowitz
   • CommentTimeNov 29th 2011
   • (edited Nov 30th 2011)
   • PermaLink
   Author: Tom Hirschowitz
   Format: MarkdownItexUsing the notations of the first post, the second universal property is that of a terminal object in the category with objects tuples of a set $x \colon 1 \to sets$, and natural transformations $a \colon x! \to F$ and $b \colon x! \to G$ such that $\alpha \circ (aL) = \beta \circ (bR)$, and morphisms $(x,a,b) \to (x',a',b')$ all transformations $\gamma \colon x \to x'$ such that $a' \circ (x!) = a$ and $b' \circ (x!) = b$.

   Using the notations of the first post, the second universal property is that of a terminal object in the category with objects tuples of a set $x \colon 1 \to sets$, and natural transformations $a \colon x! \to F$ and $b \colon x! \to G$ such that $\alpha \circ (aL) = \beta \circ (bR)$, and morphisms $(x,a,b) \to (x',a',b')$ all transformations $\gamma \colon x \to x'$ such that $a' \circ (x!) = a$ and $b' \circ (x!) = b$.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeDec 7th 2011
   • (edited Dec 7th 2011)
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexOkay. I'm sorry for being a bit snappy, and for taking so long to reply now; I had something else on my mind. But it's good, because if I had tried to answer you a week ago, I don't think I would have had the right answer, whereas now I think I see it. Suppose we have a [[2-category equipped with proarrows]] $K$. (I know this may seem like a great leap into abstraction, but trust me.) Suppose $J\colon A ⇸ B$ is a proarrow, and $f\colon A\to X$ is an arrow. Then a **limit** of $f$ weighted by $J$ is an arrow $\ell\colon B\to X$ equipped with an isomorphism $\ell_\bullet \cong f_\bullet \lhd J$. Here $\lhd$ denotes the right adjoint on one side to composition of proarrows, and $f_\bullet\colon A ⇸ X$ denotes the proarrow represented by $f$ (its right adjoint is $f^\bullet$). In particular, if $K=Cat$ with [[profunctors]] as proarrows, then: * Given $A$, take $J = t_\bullet$ for $t\colon A\to 1$ the unique arrow. Then $J$-weighted limits are ordinary limits of $A$-shaped diagrams. * Given $j\colon A \to B$, take $J = j_\bullet$; then $J$-weighted limits are ordinary (pointwise) right Kan extensions along $j$. Moreover, phrasing things this way, we have a nice functoriality property: given $J_1 \colon A ⇸ B$ and $J_2 \colon B ⇸ C$, we have $$ \mathrm{lim}^{J_1 \odot J_2} f \cong \mathrm{lim}^{J_2} \mathrm{lim}^{J_1} f.$$ Now, if $K$ is equipped with proarrows and the bicategory of proarrows has local colimits (colimits in hom-categories preserved by composition), then we can construct a new 2-category equipped with proarrows, denoted $Mod(K)$, as follows. * Its objects are categories enriched in the bicategory of proarrows for $K$. These are like "monads with many objects". They have a collection of objects $ob(C)$, each assigned to an object of $K$ as its "extent", and if $x$ and $y$ have extents $\epsilon x$ and $\epsilon y$, then the hom-object $C(x,y)$ is a proarrow $\epsilon y ⇸ \epsilon x$. There are then composition/multiplication and unit morphisms as usual. * An arrow from $C$ to $D$ consists of an assignation of an object $f(x)$ of $D$ to every object $x$ of $C$, together with an *arrow* $f_x\colon \epsilon x\to \epsilon(f(x))$ (not a proarrow) in $K$, and for each $x,y$ in $C$ morphisms $C(x,y) \odot (f_x)_\bullet \to (f_y)_\bullet \odot D(f(x),f(y))$. * A proarrow in $Mod(K)$ is a "bimodule" in the obvious sense, and the 2-cells are bimodule morphisms. If $K$ has one object, one arrow, and its proarrows are a monoidal category $V$, then $Mod(V)$ is the bicategory of $V$-enriched categories, functors, and profunctors. Note also that every object $X$ of $K$ induces an object of $Mod(K)$, call it $y X$, which has one object $x$ of extent $X$ and the unit profunctor $U_X$ as $y X(x,x)$. This is a full embedding $K\hookrightarrow Mod(K)$. Now, let's take $K=Cat$ with profunctors as the proarrows, and suppose given the data described in your first post. Then there is an object of $Mod(Cat)$, let's call it $D$, described as follows. It has three objects $a$, $b$, and $c$, with $\epsilon a = A$, $\epsilon b = B$, and $\epsilon c = C$. We have $D(b,a) = L_\bullet$, $D(c,a) = R_\bullet$, and all other hom-objects empty. Now your functors $F,G,H$ and transformations $\alpha,\beta$ describe an arrow $f\colon D\to y(set)$ in $Mod(Cat)$. I claim that the object you want is the right Kan extension of $f$ along the unique arrow $D\to 1$ in $Mod(Cat)$. Moreover, the fact that your two ways of computing this agree boils down to the functoriality of limits I described above. On the one hand, we have an object $P$ with three objects $a,b,c$ all of extent $1$ and only trivial homs. An arrow $P\to y(set)$ is exactly a cospan in $set$, and its Kan extension along $P\to 1$ in $Mod(K)$ is the pullback of this cospan. The process of "reflecting the whole diagram in $[1,set]$" I believe corresponds to taking the Kan extension along the evident arrow $D\to P$; thus the composite Kan extension is just the Kan extension along $D\to 1$. On the other hand, we also have a category $E$ which is the **collage** of $D$. This means it is the lax colimit of $D$ regarded as a lax functor into $Prof$, and it has the additional universal property that the coprojections into the colimit are representable profunctors, and *detect representability* in the sense that a profunctor out of $E$ is representable iff its composites with all the coprojections are so. Moreover, the coprojections fit together to give an arrow $q\colon D \to y(E)$ in $Mod(Cat)$ such that $q_\bullet$ is a ("Morita") equivalence. Now the arrow $f\colon D\to y(set)$ determined by $F,G,H,\alpha,\beta$ can equivalently be regarded as a lax cocone under $D$ *qua* lax functor into $Prof$, with vertex $set$. Since colimits in (lax) slice categories are just (lax) colimits of the underlying objects, the pushout of this diagram in $Cat//set$ is just $E$, and the induced functor $E\to set$ is the same as that induced by $f\colon D\to y(set)$ and the Morita equivalence $q_\bullet\colon D\simeq y(E)$ --- which is *also* the right Kan extension of $f$ along $q$. Therefore, the process of taking the pushout in $Cat//set$ and then taking the limit of the resulting $E$-diagram is equivalent to first Kan extending along $D\to y(E)$, then along $E\to 1$; thus the composite is again the Kan extension along $D\to 1$. The nLab doesn't yet do a great job of explaining the $Mod$ construction and collages, I think. Good papers to read are "Enriched categories and cohomology" and "Cauchy characterization of enriched categories" by Ross Street, and "An axiomatics for bicategories of modules" by Carboni, Kasangian, and Walters.

   Okay. I’m sorry for being a bit snappy, and for taking so long to reply now; I had something else on my mind. But it’s good, because if I had tried to answer you a week ago, I don’t think I would have had the right answer, whereas now I think I see it.

   Suppose we have a 2-category equipped with proarrows $K$. (I know this may seem like a great leap into abstraction, but trust me.) Suppose $J\colon A ⇸ B$ is a proarrow, and $f\colon A\to X$ is an arrow. Then a limit of $f$ weighted by $J$ is an arrow $\ell\colon B\to X$ equipped with an isomorphism $\ell_\bullet \cong f_\bullet \lhd J$. Here $\lhd$ denotes the right adjoint on one side to composition of proarrows, and $f_\bullet\colon A ⇸ X$ denotes the proarrow represented by $f$ (its right adjoint is $f^\bullet$).

   In particular, if $K=Cat$ with profunctors as proarrows, then:

   • Given $A$, take $J = t_\bullet$ for $t\colon A\to 1$ the unique arrow. Then $J$-weighted limits are ordinary limits of $A$-shaped diagrams.

   • Given $j\colon A \to B$, take $J = j_\bullet$; then $J$-weighted limits are ordinary (pointwise) right Kan extensions along $j$.

   Moreover, phrasing things this way, we have a nice functoriality property: given $J_1 \colon A ⇸ B$ and $J_2 \colon B ⇸ C$, we have

   $$\mathrm{lim}^{J_1 \odot J_2} f \cong \mathrm{lim}^{J_2} \mathrm{lim}^{J_1} f.$$

   Now, if $K$ is equipped with proarrows and the bicategory of proarrows has local colimits (colimits in hom-categories preserved by composition), then we can construct a new 2-category equipped with proarrows, denoted $Mod(K)$, as follows.

   • Its objects are categories enriched in the bicategory of proarrows for $K$. These are like “monads with many objects”. They have a collection of objects $ob(C)$, each assigned to an object of $K$ as its “extent”, and if $x$ and $y$ have extents $\epsilon x$ and $\epsilon y$, then the hom-object $C(x,y)$ is a proarrow $\epsilon y ⇸ \epsilon x$. There are then composition/multiplication and unit morphisms as usual.

   • An arrow from $C$ to $D$ consists of an assignation of an object $f(x)$ of $D$ to every object $x$ of $C$, together with an arrow $f_x\colon \epsilon x\to \epsilon(f(x))$ (not a proarrow) in $K$, and for each $x,y$ in $C$ morphisms $C(x,y) \odot (f_x)_\bullet \to (f_y)_\bullet \odot D(f(x),f(y))$.

   • A proarrow in $Mod(K)$ is a “bimodule” in the obvious sense, and the 2-cells are bimodule morphisms.

   If $K$ has one object, one arrow, and its proarrows are a monoidal category $V$, then $Mod(V)$ is the bicategory of $V$-enriched categories, functors, and profunctors.

   Note also that every object $X$ of $K$ induces an object of $Mod(K)$, call it $y X$, which has one object $x$ of extent $X$ and the unit profunctor $U_X$ as $y X(x,x)$. This is a full embedding $K\hookrightarrow Mod(K)$.

   Now, let’s take $K=Cat$ with profunctors as the proarrows, and suppose given the data described in your first post. Then there is an object of $Mod(Cat)$, let’s call it $D$, described as follows. It has three objects $a$, $b$, and $c$, with $\epsilon a = A$, $\epsilon b = B$, and $\epsilon c = C$. We have $D(b,a) = L_\bullet$, $D(c,a) = R_\bullet$, and all other hom-objects empty.

   Now your functors $F,G,H$ and transformations $\alpha,\beta$ describe an arrow $f\colon D\to y(set)$ in $Mod(Cat)$. I claim that the object you want is the right Kan extension of $f$ along the unique arrow $D\to 1$ in $Mod(Cat)$. Moreover, the fact that your two ways of computing this agree boils down to the functoriality of limits I described above.

   On the one hand, we have an object $P$ with three objects $a,b,c$ all of extent $1$ and only trivial homs. An arrow $P\to y(set)$ is exactly a cospan in $set$, and its Kan extension along $P\to 1$ in $Mod(K)$ is the pullback of this cospan. The process of “reflecting the whole diagram in $[1,set]$” I believe corresponds to taking the Kan extension along the evident arrow $D\to P$; thus the composite Kan extension is just the Kan extension along $D\to 1$.

   On the other hand, we also have a category $E$ which is the collage of $D$. This means it is the lax colimit of $D$ regarded as a lax functor into $Prof$, and it has the additional universal property that the coprojections into the colimit are representable profunctors, and detect representability in the sense that a profunctor out of $E$ is representable iff its composites with all the coprojections are so. Moreover, the coprojections fit together to give an arrow $q\colon D \to y(E)$ in $Mod(Cat)$ such that $q_\bullet$ is a (“Morita”) equivalence.

   Now the arrow $f\colon D\to y(set)$ determined by $F,G,H,\alpha,\beta$ can equivalently be regarded as a lax cocone under $D$ qua lax functor into $Prof$, with vertex $set$. Since colimits in (lax) slice categories are just (lax) colimits of the underlying objects, the pushout of this diagram in $Cat//set$ is just $E$, and the induced functor $E\to set$ is the same as that induced by $f\colon D\to y(set)$ and the Morita equivalence $q_\bullet\colon D\simeq y(E)$ — which is also the right Kan extension of $f$ along $q$.

   Therefore, the process of taking the pushout in $Cat//set$ and then taking the limit of the resulting $E$-diagram is equivalent to first Kan extending along $D\to y(E)$, then along $E\to 1$; thus the composite is again the Kan extension along $D\to 1$.

   The nLab doesn’t yet do a great job of explaining the $Mod$ construction and collages, I think. Good papers to read are “Enriched categories and cohomology” and “Cauchy characterization of enriched categories” by Ross Street, and “An axiomatics for bicategories of modules” by Carboni, Kasangian, and Walters.

9. • CommentRowNumber9.
   • CommentAuthorTom Hirschowitz
   • CommentTimeDec 10th 2011
   • PermaLink
   Author: Tom Hirschowitz
   Format: TextThis looks like exactly the kind of things I was hoping for: thanks a lot! The papers you point to look very frightening; do they require prior knowledge of proarrow equipment (or your framed bicategories)?

   This looks like exactly the kind of things I was hoping for: thanks a lot! The papers you point to look very frightening; do they require prior knowledge of proarrow equipment (or your framed bicategories)?
10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeDec 11th 2011
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexNo, they don't; those papers are written entirely in terms of bicategories, with functors considered as the profunctors with right adjoints. This simplifies things somewhat (at the price of making it impossible to distinguish between a category and its Cauchy completion).

    No, they don’t; those papers are written entirely in terms of bicategories, with functors considered as the profunctors with right adjoints. This simplifies things somewhat (at the price of making it impossible to distinguish between a category and its Cauchy completion).

11. • CommentRowNumber11.
    • CommentAuthorTom Hirschowitz
    • CommentTimeDec 20th 2011
    • (edited Dec 20th 2011)
    • PermaLink
    Author: Tom Hirschowitz
    Format: MarkdownItexDear Mike, Thanks, I'm almost there! However, I'm clearly not yet autonomous in the use of these tools, and you're probably not ok for being systematically used as a reference manual. So is there some place I could look up for known facts on these things? E.g., I'd expect the following to hold: * $Mod(Prof)$ has all right extensions (but how are they computed?), * so, starting from the original $L,R,F,G,H$, and $\alpha$, $\beta$, * lett $E$ be the pushout of $L$ and $R$ in $Cat$, * see it as a one-object Prof-category, * the pushout square is a map $D \to E$ in $Mod(Cat)$; * thus, successively extending along $D \to E \to 1$, we again obtain the same limit as before.

    Dear Mike,

    Thanks, I’m almost there! However, I’m clearly not yet autonomous in the use of these tools, and you’re probably not ok for being systematically used as a reference manual. So is there some place I could look up for known facts on these things?

    E.g., I’d expect the following to hold:

    • $Mod(Prof)$ has all right extensions (but how are they computed?),

    • so, starting from the original $L,R,F,G,H$, and $\alpha$, $\beta$,

      • lett $E$ be the pushout of $L$ and $R$ in $Cat$,
      • see it as a one-object Prof-category,
      • the pushout square is a map $D \to E$ in $Mod(Cat)$;
      • thus, successively extending along $D \to E \to 1$, we again obtain the same limit as before.
12. • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeDec 20th 2011
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexIt may be faster for me to answer your questions directly than to try to track down references; there isn't a good textbook or anything on profunctor theory (sadly). For any bicategory $B$ with sufficient limits in its hom-categories, the bicategory $Mod(B)$ has right extensions and right liftings, computed in basically the same way as right extensions and liftings are in $Prof$ using the right extensions in $B$ and an end-like limit. But perhaps what you wanted to ask about was right extensions of *functors*, which is the sense in which we "extend along $D\to E\to 1$". Those are a limit-notion and depend on completeness properties of the target object — in your case, $set$. I don't know offhand a general result about how completeness properties of an object of a bicategory/equipment $B$ extend to its image in $Mod(B)$, but it should be provable.

    It may be faster for me to answer your questions directly than to try to track down references; there isn’t a good textbook or anything on profunctor theory (sadly). For any bicategory $B$ with sufficient limits in its hom-categories, the bicategory $Mod(B)$ has right extensions and right liftings, computed in basically the same way as right extensions and liftings are in $Prof$ using the right extensions in $B$ and an end-like limit. But perhaps what you wanted to ask about was right extensions of functors, which is the sense in which we “extend along $D\to E\to 1$”. Those are a limit-notion and depend on completeness properties of the target object — in your case, $set$. I don’t know offhand a general result about how completeness properties of an object of a bicategory/equipment $B$ extend to its image in $Mod(B)$, but it should be provable.