The Kan extension of a functor $F : C \to D$ with respect to a functor
is, if it exists, a kind of best approximation to the problem of finding a functor $C' \to D$ such that
hence to extending the domain of $F$ through $p$ from $C$ to $C'$.
More generally, this makes sense not only in Cat but in any 2-category.
Similarly, a Kan lift is the best approximation to lifting a morphism $F : C \to D$ through a morphism
to a morphism $\hat F$
Kan extensions are ubiquitous. See the discussion at Examples below.
There are various slight variants of the definition of Kan extension . In good cases they all exist and all coincide, but in some cases only some of these will actually exist.
We (have to) distinguish the following cases:
“ordinary” or “weak” Kan extensions
These define the extension of an entire functor, by an adjointness relation.
Here we (have to) distinguish further between
which define extensions of all possible functors of given domain and codomain (if all of them indeed exist);
which define extensions of single functors only, which may exist even if not every functor has an extension.
“pointwise” or “strong” Kan extensions
These define the value of an extended functor on each object (each “point”) by a weighted (co)limit.
Furthermore, a pointwise Kan extension can be “absolute”.
If the pointwise version exists, then it coincides with the “ordinary” or “weak” version, but the former may exist without the pointwise version existing. See below for more.
Some authors (such as Kelly) assert that only pointwise Kan extensions deserve the name “Kan extension,” and use the term as “weak Kan extension” for a functor equipped with a universal natural transformation. It is certainly true that most Kan extensions which arise in practice are pointwise. This distinction is even more important in enriched category theory.
Let
be a functor. For $D$ any other category, write
for the induced functor on the functor categories: this sends a functor $h : C' \to D$ to the composite functor $p^* h : C \stackrel{p}{\to} C' \stackrel{h}{\to} D$.
If $p^*$ has a left adjoint, typically denoted
or
then this left adjoint is called the ( ordinary or weak ) left Kan extension operation along $p$. For $h \in [C,D]$ we call $p_! h$ the left Kan extension of $h$ along $p$.
Similarly, if $p^*$ has a right adjoint, this right adjoint is called the right Kan extension operation along $p$. It is typically denoted
or
The analogous definition clearly makes sense as stated in other contexts, such as in enriched category theory.
If $C' = *$ is the terminal category, then
The functor $p^*$ in this case sends objects $d$ of $D$ to the constant functor $\Delta_d$ on $d$. Notice that for $F \in [C,D]$ any functor,
a natural transformation $\Delta_d \to F$ is the same as a cone over $F$;
a natural transformation $F \to \Delta_d$ is the same as a cocone under $F$.
Therefore the natural hom-isomorphisms of the adjoint functors $(p_! \dashv p^*)$ and $(p^* \dashv p_*)$
and
assert that
There is also a local definition of “the Kan extension of a given functor $F$ along $p$” which can exist even if the entire functor defined above does not. This is a generalization of the fact that a particular diagram of shape $C$ can have a limit even if not every such diagram does. It is also a special case of the fact discussed at adjoint functor that an adjoint functor can fail to exist completely, but may still be partially defined. If the local Kan extension of every single functor exists for some given $p\colon C\to C'$ and $D$, then these local Kan extensions fit together to define a functor which is the global Kan extension.
Thus, by the general notion of “partial adjoints”; we say
The local left Kan extension of a functor $F\in [C,D]$ along $p : C \to C'$* is, if it exists, a functor
equipped with a natural isomorphism
hence a (co)representation of the functor $Hom_{[C,D]}(F,p^*(-))$.
The local definition of right Kan extensions along $p$ is dual.
As for adjoints and limits, by the usual logic of representable functors this can equivalently be rephrased in terms of universal morphisms:
The left Kan extension $Lan F = Lan_p F$ of $F : C \to D$ along $p :C\to C'$ is a functor $Lan F : C' \to D$ equipped with a natural transformation $\eta_F : F \Rightarrow p^* Lan F$.
with the property that every other natural transformation $F \Rightarrow p^* G$ factors uniquely through $\eta_F$ as
Similarly for the right Kan extension, with the direction of the natural transformations reversed:
By the usual reasoning (see e.g. Categories Work, chapter IV, theorem 2), if these representations exist for every $F$ then they can be organised into a left (right) adjoint $Lan_p$ ($Ran_p$) to $p^*$.
The definition in this form makes sense not just in Cat but in every 2-category. In slightly different terminology, the left Kan extension of a 1-cell $F:C\to D$ along a 1-cell $p\in K(C,C')$ in a 2-category $K$ is a pair $(Lan_p F,\alpha)$ where $\alpha : F\to Lan_p F\circ p$ is a 2-cell which reflects the object $F\in K(C,D)$ along the functor $p^* = K(p,D):K(C',D)\to K(C,D)$. Equivalently, it is such a pair such that for every $G\colon C' \to D$, the function
is a bijection.
In this form, the definition generalizes easily to any n-category for any $n\ge 2$. If $K$ is an $n$-category, we say that the left Kan extension of a 1-morphism $F:C\to D$ along a 1-morphism $p\in K(C,C')$ is a pair $(Lan_p F,\alpha)$, where $Lan_p F \colon C' \to D$ is a 1-morphism and $\alpha : F\to Lan_p F\circ p$ is a 2-morphism, with the property that for any 1-morphism $G\colon C'\to D$, the induced functor
is an equivalence of $(n-2)$-categories.
We say that a Kan extension $Lan_p F$ is preserved by a functor $G$ if the composite $G \circ Lan_p F$ is a Kan extension of $G F$ along $p$, and moreover the universal natural transformation $G F \to G(Lan_p F)p$ is the composite of $G$ with the universal transformation $F\to (Lan_p F)p$.
If the codomain category $D$ admits certain (co)limits, then left and right Kan extensions can be constructed, over each object (“point”) of the domain category $C'$ out of these: Kan extensions that admit this form are called pointwise. (Reviews include (Riehl, I 1.3)).
The notion of pointwise Kan extensions deserves to be discussed in the general context of enriched category theory, which we do below. The reader may want to skip ahead to the section
which discusses the situation in ordinary (Set-enriched) category theory in terms of ordinary limits (“conical” limits, defined in terms of cones, to be distinguished from the more general weighted limits). While the formulas in that case are classical and fundamentally useful in practice, they do rely heavily on special properties of the enriching category Set.
The general formulation of pointwise Kan extensions in general enriched contexts is
In the case that the codomain category is (co)tensored these may be expressed equivalently
First, here is a characterization that doesn’t rely on any computational framework:
A Kan extension, def. 2, is called pointwise if and only if it is preserved by all representable functors.
(Categories Work, theorem X.5.3)
Suppose given $F : C \to D$ and $p : C \to C'$ such that for every $c' \in C'$, the weighted limit
exists. Then these objects fit together into a functor $Ran_p F$ which is a right Kan extension of $F$ along $p$. Dually, if the weighted colimit
exists for all $c'$, then they fit together into a left Kan extension $Lan_p F$. These definitions evidently make sense in the generality of $V$-enriched category theory for $V$ a closed symmetric monoidal category. (In fact, they can be modified slightly to make sense in the full generality of a 2-category equipped with proarrows.)
In particular, this means that if $C$ is small and $D$ is complete (resp. cocomplete), then all right (resp. left) Kan extensions of functors $F\colon C\to D$ exist along any functor $p\colon C\to C'$.
One can prove that any Kan extension constructed in this way must be pointwise, in the sense of being preserved by all representables as above. Moreover, conversely, if a Kan extension $Lan_p F$ is pointwise, then one can prove that $(Lan_p F)(c')$ must be in fact a $C'(p(-),c')$-weighted colimit of $F$, and dually; thus the two notions are equivalent.
If the $V$-enriched category $D$ is powered over $V$, then the above weighted limit may be re-expressed in terms of an end as
So in particular when $D = V$ this is
Similarly, if $D$ is tensored over $V$, then the left Kan extension is given by a coend.
The coend formula for the left Kan extension is nicely understood when thinking of $C$ and $D$ above as opposite categories and for $\mathcal{V} = Set$, so that it takes presheaves $F$ on $C$ along $p \colon C \to C'$ to presheaves $Lan_p F$ on $C'$, by the formula
Using the Yoneda lemma to rewrite $F(c) \simeq Hom_{PSh(C)}(c,F)$, this is
In this form one sees that what the coend does is to produce the set whose elements are equivalence classes of pairs of morphisms
where two such are regarded as equivalent whenever there is $f \colon c'_1 \to c'_2$ such that
This is particularly suggestive in cases when we may think of the objects of $C$ and $C'$ on the same footing, notably when $p$ is a full subcategory inclusion. For in that case we may imagine that a representative pair $(c' \to p(c), c \to F)$ is a stand-in for the actual pullback of elements of $F$ via forming the composite “$c'\to c \to F$”, only that this composite is not defined. But the above equivalence relation is precisely that under which this composite would be invariant.
In the case of functors between ordinary locally small categories, hence in the special case of $V$-enriched category theory for $V =$ Set, there is an expression of a weighted (co)limit and hence a pointwise Kan extension as an ordinary (“conical”, meaning: in terms of cones) (co)limit over a comma category:
Let
$C$ be a small category;
$D$ have all small limits.
Then the right Kan extension of a functor $F : C \to D$ of locally small categories along a functor $p : C \to C'$ exists and its value on an object $c' \in C'$ is given by the limit
where
$\Delta_{c'}/p$ is the comma category;
$\Delta_{c'}/p \to C$ is the canonical forgetful functor.
Likewise, if $D$ admits small colimits, the left Kan extension of a functor exists and is pointwise given by the colimit
This appears for instance as (Borceux, I, thm 3.7.2). Discussion in the context of enriched category theory is in (Kelly, section 3.4).
A cartoon picture of the forgetful functor out of the comma category $p/\Delta_{c'} \to C$, useful to keep in mind, is
The comma category here is equivalently the category of elements of the functor $C'(p(-), c') : C^{op} \to Set$
Consider the case of the left Kan extension, the other case works analogously, but dually.
First notice that the above pointwise definition of values of a functor canonically extends to an actual functor:
for $\phi : c'_1 \to c'_2$ any morphism in $C'$ we get a functor
of comma categories, by postcomposition. This morphism of diagrams induces canonically a corresponding morphism of colimits
Now for the universal property of the functor $Lan_p F$ defined this way. For $c' \in C'$ denote the components of the colimiting cocone $(Lan_p F)(c') := \lim_{\to}( p/\Delta_{c'} \to C \stackrel{F}{\to} D)$ by $s_{(-)}$, as in
We now construct in components a natural transformation
for $Lan_p F$ defined as above, and show that it satisfies the required universal property. The components of $\eta_F$ over $c \in C$ are morphisms
Take these to be given by
(this is similar to what happens in the proof of the Yoneda lemma, all of these arguments are variants of the argument for the Yoneda lemma, and vice versa). It is straightforward, if somewhat tedious, to check that these are natural, and that the natural transformation defined this way has the required universal property.
We have seen that if $D$ has enough limits or colimits, then a pointwise Kan extension can be defined in terms of these limits, and will necessarily satisfy the universal property described first. However, not all Kan extensions are pointwise: that is, having a universal transformation $F \to (Lan_p F)p$ does not necessarily imply that the individual values of $Lan_p F$ are limits or colimits in its codomain. Non-pointwise Kan extensions can exist even when $D$ does not admit very many limits.
It should be noted, though, that pointwise Kan extensions can still exist, and hence the particular requisite limits/colimits exist, even if $D$ is not (co)complete. For instance, the Kan extensions that arise in the study of derived functors are pointwise, and in fact absolute (preserved by all functors), even though their codomains are homotopy categories which generally do not admit all limits and colimits.
Non-pointwise Kan extensions seem to be very rare in practice. However, the abstract notion of Kan extension (sometimes called simply “extension”) in a 2-category, and its dual notion of lifting, can be useful in 2-category theory. For instance, bicategories such as Prof admit all right extensions and right liftings; a bicategory with this property may be considered a horizontal categorification of a closed monoidal category.
Am absolute Kan extension $Lan_p F$ is one which is preserved by all functors $G$ out of the codomain of $F$:
(same for right Kan extensions).
The most prominent example of absolute Kan extensions is given by adjoint functors; in fact they can be defined as certain absolute Kan extensions. See there for the precise statement.
Absolute Kan extensions are always pointwise, as the latter can be defined as those preserved by representables; there are (lots of) examples of pointwise Kan extensions which are not absolute.
Note that in a general 2-category, absolute Kan extensions make perfect sense, while for defining pointwise ones more structure is needed: comma objects and/or some structure which would let us work with (co)limits inside that 2-category (such as a (co)Yoneda structure? or a proarrow equipment).
The global definition of Kan extensions for functors in terms of left/right adjoints to pullbacks may be interpreted essentially verbatim in the context of (∞,1)-categories
See at (∞,1)-Kan extension.
The Kan extension of a functor may be regarded more abstractly as an extension-problem in the 2-category Cat of categories. The same extension problem can be stated verbatim in any 2-category and hence there is a corresponding more general notion of Kan extensions of 1-morphisms in 2-categories. This is discussed in (Lack 09, section 2.2).
The question of defining a pointwise Kan extension in a general 2-category is more subtle, and there are at least two distinct approaches. If the 2-category has comma objects, then we can define a Kan extension to be pointwise if it remains a Kan extension upon pasting with any comma object; this is an “internalization” of the above definition in terms of conical colimits. On the other hand, in a 2-category equipped with proarrows we can define pointwise Kan extensions as particular weighted (co)limits using a representable weight; this generalizes the above definition as a weighted (co)limit.
In some 2-categories such as $Cat$, both definitions agree; but in others they do not, and in general in this case it is the equipment-theoretic version that is “correct”. For instance, in $V Cat$ the equipment-theoretic version gives the right notion of pointwise Kan extension, whereas the comma-object one is too strong.
As a concrete example, let $V=Cat$, so that $V Cat = 2 Cat$; then comma objects are not informative enough because they “don’t see the 2-cells”. In even more specificity, let $B$ be the walking 2-cell and $M$ the walking pair of parallel 1-morphisms, with $f:1\to B$ and $g:1\to M$ the inclusions of the common domain of the parallel 1-morphisms; then the equipment-theoretic-pointwise $Lan_f g$ is constant at the domain object, whereas the comma-object-pointwise $Lan_f g$ does not exist. See (Roald, Example 2.24) for details.
Let $\mathcal{V}$ be a suitable enriching category (a cosmos). Notably $\mathcal{V}$ may be Set.
For $F : C \to D$ a $\mathcal{V}$-enriched functor between small $\mathcal{V}$-enriched categories we have
the left Kan extension along $F$ takes representable presheaves $C(c,-) : C \to \mathcal{V}$ to their image under $F$:
for all $c \in C$.
if $F$ is a full and faithful functor then $F^* (Lan_F H) \simeq H$ and in fact the $(Lan_F \dashv F^*)$-unit of an adjunction is a natural isomorphism
whence it follows (see the basic properties of adjoint functors) that $Lan_F : [C,\mathcal{V}] \to [D,\mathcal{V}]$ is itself a full and faithful functor.
The second statement appears for instance as (Kelly, prop. 4.23).
For the first statement, using the coend formula for the left Kan extension above we have naturally in $d' \in D$ the expression
Here the last step is called sometimes the co-Yoneda lemma. It follows for instance by observing that $\int^{c' \in C} D(F(c'), d') \cdot C(c,c')$ is equivalently dually the expression for the left Kan extension of the non-representable $D(F(-),d') : C^{op} \to \mathcal{V}$ along the identity functor.
Similarly for the second, if $H : D \to E$ is any $\mathcal{V}$-enriched functor with $E$ tensored over $\mathcal{V}$, then its left Kan extension evaluated on the image of $F$ is
The following statement says that left exact functors into toposes have left exact left Kan extension along the Yoneda embedding (Yoneda extension) and that this is the inverse image of a geometric morphism of sheaf toposes if the original functor preserves covers.
(We state this in (∞,1)-category theory, the same statement holds true in plain category theory by just disregarding all occurences of “$\infty$”.)
Let $\mathbf{H}$ be an (∞,1)-topos and let $\mathcal{C}$ be an (∞,1)-site with (∞,1)-sheaf (∞,1)-category $Sh(\mathcal{C})$. Then the (∞,1)-functor
given by precomposition with ∞-stackification/sheafification $L$ and with the (∞,1)-Yoneda embedding $Y$ is a full and faithful (∞,1)-functor. Moreover, its essential image consisist of those (∞,1)-functors $f \colon \mathcal{C} \longrightarrow \mathcal{X}$ which are left exact and which preserve covers in that for $\{U_i \to X\}_i$ a covering in $\mathcal{C}$, then $\coprod_i f(U_i) \to f(X)$ is an effective epimorphism in $\mathcal{X}$.
This appears as Lurie, HTT, prop. 6.2.3.20.
Prop. 4 is a central statement in the theory of classifying toposes. See there for more.
For more discussion of left exactness properties preserved by left Kan extension see also (Borceux-Day, Karazeris-Protsonis).
Let $f : C \to D$ be a small opfibration? of categories, and let $\mathcal{C}$ be a category with all small colimits. Then for each $d \in D$ the inclusion
of the fiber over $d$ into the comma category given by
has a left adjoint, given by
Therefore (by the discussion here) it is a cofinal functor. Accordingly, the local formula for the left Kan extension
is equivalently given by taking the colimit over the fiber:
The central point about examples of Kan extensions is:
Kan extensions are ubiquitous .
To a fair extent, category theory is all about Kan extensions and the other universal constructions: limits, adjoint functors, representable functors, which are all special cases of Kan extensions – and Kan extensions are special cases of these.
Listing examples of Kan extensions in category theory is much like listing examples of integrals in analysis: one can and does fill books with these. (In fact, that analogy has more to it than meets the casual eye: see coend for more).
Keeping that in mind, we do list some special cases and special classes of examples that are useful to know. But any list is necessarily wildly incomplete.
For $C' =$ the point, the right Kan extension of $F$ is the limit of $F$, $Ran F \simeq \lim F$ and the left Kan extension is the colimit $Lan F \simeq colim F$.
For $f : X \to Y$ a morphism of sites coming from a functor $f^t : S_Y \to S_X$ of the underlying categories, the left Kan extension of functors along $f^t$ is the inverse image operation $f^{-1} : PSh(Y) \to PSh(X)$.
see also examples of Kan extensions
We discuss examples of Kan extensions that are not point-wise
For more on the following see also
The basic example for left Kan extensions using the above pointwise formula, is in the construction of the pullback of sheaves along a morphism of topological spaces. Let $f:X\to Y$ be a continuous map and $F$ a presheaf over $X$. Then the formula $(f_* F)(U) = F(f^{-1}(U))$ clearly defines a presheaf $f_* F$ on $Y$, which is in fact a sheaf if $F$ is. On the other hand, given a presheaf $G$ over $Y$ we can not define pullback presheaf $(f^{-1} G)(V)=G(f(V))$ because $f(V)$ might not be open in general (unless $f$ is an open map). For Grothendieck sites such $f(V)$ would not make even sense. But one can consider approximating from above by $G(W)$ for all $W\supset f(V)$ which are open and take a colimit of this diagram of inclusions (all $W$ are bigger, so getting down to the lower bound means going reverse to the direction of inclusions). But inclusion $f(V)\subset W$ implies $V\subset f^{-1}(f(V))\subset f^{-1}(W)$. The latter identity $V\subset f^{-1}(W)$ involves only open sets. Thus we take a colimit over the comma category $(V\downarrow f^{-1})$ of $G$. If $G$ is a sheaf, the colimit $G(V)$ understood as a rule $V\mapsto G(V)$ is still not a sheaf, we need to sheafify. The result is sheaf-theoretic pullback
which is a sheaf, and one can analyze this construction to show that $f^{-1}$ is a left adjoint to $f_*$. This usage of left Kan extension persists in the more general case of Grothendieck topologies.
We list here some occurrences of Kan extensions in physics.
Notice that since, by the above discussion, Kan extensions are ubiquitous in category theory and are essentially equivalent to other standard universal constructions such as notably co/limits, to the extent that there is a relation between category theory and physics at all, it necessarily also involves Kan extensions, in some guise. But here is a list of some example where they appear rather explicitly.
In extended quantum field theory on open and closed manifolds, usually the theory “in the bulk” (on closed manifolds) is induced by “extending” that “on the boundary”, and in good cases this extension is explicitly a (homotopy)-Kan extension. This is the case notably for 2d TQFT in the form of TCFT (Costello 04), see at TCFT – Classification for details.
When path integral quantization is formalized in terms of fiber integration in generalized cohomology (as surveyed at motivic quantization) then the push-forward step, hence the path integral itself, is given by left homotopy Kan extension of parameterized spectra. For explicit details see (Nuiten 13, section 4.1), also (Schreiber 14, section 6.2). By example 6.3, a special case of this is the integration formulas via Kan extension in (Hopkins-Lurie 14, section 4).
Generally, for $p : C \to C'$ a functor, the induced “precomposition” functor on functor categories
is spoken of as pulling back a functor on $C'$ to a functor on $C$, as this operation goes in the direction opposite to that of $p$ itself. For this reason, we have above denoted this functor by $p^*$. Likewise, one might call the (left or right) Kan extensions along $p$ a push forward of functors from $C$ to functors on $C'$.
This notation also coincides with that for geometric morphisms in one case: any functor $p\colon C\to C'$ between small categories induces a geometric morphism $[C,Set] \to [C',Set]$ of presheaf toposes, whose inverse image is the above $p^*$ and whose direct image $p_*$ is the right Kan extension functor. Note that $p^*$ preserves (finite) limits, as required of an inverse image functor, since it has a left adjoint, namely left Kan extension.
On the other hand, if $p$ is additionally a flat functor, then the above precomposition functor is also the direct image of a geometric morphism, whose inverse image is given by left Kan extension (which preserves finite limits when $p$ is flat). More generally, if $C^{op}$ and $(C')^{op}$ are sites and $p^{op}\colon C^{op}\to (C')^{op}$ is flat and preserves covering families (i.e. it is a morphism of sites), then precomposition is the direct image of a geometric morphism $Sh(C^{op})\to Sh((C')^{op})$ between sheaf toposes.
For example, $C^{op}$ and $(C')^{op}$ might be the posets $Open(X)$ and $Open(X')$ of open subsets of topological spaces (or locales) $X$ and $X'$ and inclusions, in which case
come from continuous maps of topological spaces going the other way
via the usual inverse image $f^{-1} : O(X)^{op} \to O(X')^{op}$ of open subsets.
Thus, in such cases, the functor $p^*$, which looks like a pullback of functors along $p = f^{-1}$, corresponds geometrically to a push-forward of (pre)sheaves along $f$. Therefore, in presheaf literature (such as Categories and Sheaves) the precomposition functor induced by $p$ is usually denoted $p_*$ and not $p^*$.
It is however noteworthy that also the opposite perspective does occur in geometrically motivated examples. For instance
if $C$ is the discrete category on smooth space and $D = U(1)$ is the discrete category on the smooth space $X$ underlying the Lie group $U(1)$, then smooth functors (i.e. functors internal to smooth spaces) $F : C \to D$ can be identified with smooth $U(1)$-valued functions on $X$, and the functor on these functor categories induced by a smooth functor $p : C \to C'$ does correspond to the familiar notion of pullback of functions;
and similar in higher degrees: if $C = P_1(X)$ is the smooth path groupoid of a smooth space and $D = \mathbf{B} U(1)$ the smooth group $U(1)$ regarded as a one-object Lie groupoid, then smooth functors $C \to D$ correspond to smooth 1-forms $\in \Omega^1(X)$ on $X$, and precomposition with a smooth functor $p : P_1(X) \to P_1(X')$ corresponds to the familiar notion of pullback of 1-forms.
This means that whether or not Kan extensions correspond geometrically to pushforward or to pullback depends on the way (covariant or contravariant) in which the domain categories $C$, $C'$ are identified with geometric entities.
Kan extension, (∞,1)-Kan extension
Textbook sources include
Francis Borceux, section 3.7 of Handbook of Categorical Algebra I
Kashiwara and Shapira, section 2.3 in Categories and Sheaves
The book
has a famous treatment of Kan extensions with a statement: “The notion of Kan extensions subsumes all the other fundamental concepts in category theory”. Of course, many other fundamental concepts of category theory can also be regarded as subsuming all the others.
Lecture notes with an eye towards applications in homotopy theory include
For Kan extensions in the context of enriched category theory see
and chapter 4 of
The (∞,1)-category theory notion is discussed in section 4.3 of
For uses of Kan extension in the study of algebras over an algebraic theory see
Preservation of certain limits by left Kan extended functors is discussed in
Francis Borceux, and Brian Day, On product-preserving Kan extension, Bulletin of the Australian Mathematical Society, Vol 17 (1977), 247-255 (pdf)
Panagis Karazeris, and Grigoris Protsonis, Left Kan extensions preserving finite products (pdf)
The general notion of extensions of 1-morphisms in 2-categories is discussed in