Monads and tensor products
M. Zarichnyi defined an operation of tensor product for each functor that can be complemented to a monad. We investigate the existence of tensor product for functors which cannot be complemented to monads.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507590496092160 |
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| author | Radul, T. N. Радул, Т. Н. |
| author_facet | Radul, T. N. Радул, Т. Н. |
| author_sort | Radul, T. N. |
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| description | M. Zarichnyi defined an operation of tensor product for each functor that can be complemented to a monad. We investigate
the existence of tensor product for functors which cannot be complemented to monads. |
| first_indexed | 2026-03-24T02:11:44Z |
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UDC 515.12
T. Radul (Inst. Math., Casimirus Great Univ., Bydgoszcz, Poland and Lviv Nat. I. Franko Univ.)
MONADS AND TENSOR PRODUCTS
МОНАДИ TA ТЕНЗОРНI ДОБУТКИ
M. Zarichnyi defined an operation of tensor product for each functor that can be complemented to a monad. We investigate
the existence of tensor product for functors which cannot be complemented to monads.
М. Зарiчний означив операцiю тензорного добутку для кожного функтора, що доповнюється до монади. У цiй
статтi дослiджено iснування тензорного добутку для функторiв, якi не можна доповнити до монади.
0. Introduction. The general theory of functors acting on the category \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} of compact Hausdorff
spaces (compacta) and continuous mappings was founded by Shchepin [9]. He described some
elementary properties of such functors and defined the notion of the normal functor which has
become very fruitful. The classes of all normal functors include many classical constructions: the
hyperspace exp, the space of probability measures P, the space of idempotent measures I, and many
other functors (see [4, 10, 11]).
The algebraic aspect of the theory of functors in categories of topological spaces and continuous
maps is based, mainly, on the existence of monad (or triple) structure in the sense of S. Eilenberg and
J. Moore [2]. This notion turned out, in particular, to be a fruitful tool for investigation of functors
in the category \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} (see [8, 10]).
We recall the definition of monad only for the category \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}. A monad \BbbT = (T, \eta , \mu ) in
the category \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} consists of an endofunctor T : \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} \rightarrow \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} and natural transformations \eta :
\mathrm{I}\mathrm{d}Comp \rightarrow T (unity), \mu : T 2 \rightarrow T (multiplication) satisfying the relations \mu \circ T\eta = \mu \circ \eta T =1T and
\mu \circ \mu T = \mu \circ T\mu . (By \mathrm{I}\mathrm{d}Comp we denote the identity functor on the category \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} and T 2 is the
superposition T \circ T of T.)
The tensor product operation of probability measures is well known and very useful for investiga-
tion of the functor P (see, for example, [3], Chapter 8). Zarichnyi has generalized the tensor product
of probability measures for any functor which can be completed to a monad [10] (Chapter 3.4). The
definition of a functor which admits a tensor product is given in [1].
Definition 0.1 [1]. We say that a functor F : \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} \rightarrow \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} admits a tensor product if for
each family of compacta \{ X\alpha \} there exists a continuous map \otimes \{ X\alpha \} :
\prod
\alpha
FX\alpha \rightarrow F
\Bigl( \prod
\alpha
X\alpha
\Bigr)
which is natural with respect to each argument and for each \alpha we have F (p\alpha )\circ \otimes = \mathrm{p}\mathrm{r}\alpha , where p\alpha :\prod
\beta
X\beta \rightarrow X\alpha and \mathrm{p}\mathrm{r}\alpha :
\prod
\beta
FX\beta \rightarrow FX\alpha are natural projections.
\Bigl(
By naturality with respect
to each argument we mean the following property: for a family of maps \{ f\alpha : X\alpha \rightarrow Y\alpha \} we have
\otimes \{ Y\alpha \} \circ
\prod
\alpha
F (f\alpha ) = F
\Bigl( \prod
\alpha
f\alpha
\Bigr)
\circ \otimes \{ X\alpha \} .
\Bigr)
The question naturally arises whether there exists a functor which can not be completed to a monad
but admits a tensor product. Firstly, we investigate the Hartman – Mycielski functor introduced in
[7]. Recently it was proved in [6] that this functor can not be completed to a monad. Unfortunately
the Hartman – Mycielski functor neither admits a tensor product, what we will prove in Section 1 of
c\bigcirc T. RADUL, 2017
854 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
MONADS AND TENSOR PRODUCTS 855
this paper. In Section 2 we investigate iterated functors. We will prove that the second iteration of a
functor, which admits a tensor product, admits tensor product too. We also will show that the second
iteration of hyperspace functor (which is functorial part of hyperspace monad, hence admits a tensor
product) can not be completed to a monad.
1. Hartman – Mycielski functor does not admit a tensor product. All functors we consider are
endofunctors in \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}. A functor F is called continuous if it preserves the limits of inverse systems.
A functor is called monomorphic if it preserves topological embeddings. For monomorphic functor
F and an embedding i : A \rightarrow X we shall identify the space F (A) and the subspace F (i)(F (A)) \subset
\subset F (X). For a functor F which preserves monomorphisms the intersection-preserving property is
defined as follows: F (\cap \{ X\alpha | \alpha \in \scrA \} ) = \cap \{ F (X\alpha ) | \alpha \in \scrA \} for every family \{ X\alpha | \alpha \in \scrA \} of
closed subsets of X. A functor is called seminormal iff it is continuous, monomorphic, preserves
empty space, one-point spaces and intersection. In what follows, all functors are assumed to be
seminormal. For such a functor there exists a unique natural transformation \eta : \mathrm{I}\mathrm{d}Comp \rightarrow F.
Each component \eta X : X \rightarrow FX is an embedding [3]. Let us remark that the map \eta X has the
following property: for each x \in X we have \eta X(x) = Fi(F\{ x\} ), where i : \{ x\} \rightarrow X is the natural
embedding.
Let X be a space and d is an admissible metric on X bounded by 1. By HM(X) we shall
denote the space of all maps from [0, 1) to the space X such that f | [ti, ti+1) \equiv const, for some
0 = t0 \leq . . . \leq tn = 1, with respect to the following metric:
dHM (f, g) =
1\int
0
d(f(t), g(t))dt, f, g \in HM(X).
The construction of HM(X) is known as the Hartman – Mycielski construction [5]. This con-
struction was considered for any compactum Z in [10] (2.5.2). Let \scrU be the unique uniformity of
Z. For every U \in \scrU and \varepsilon > 0, let
\langle \alpha ,U, \varepsilon \rangle = \{ \beta \in HM(Z) | m\{ t \in [0, 1) | (\alpha (t), \beta (t)) /\in U\} < \varepsilon \}
(here m is the Lebesgue measure on [0, 1]) The sets \langle \alpha ,U, \varepsilon \rangle form a base of a topology in HMZ.
The construction HM acts also on maps. Given a map f : X \rightarrow Y in \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}, define a map
HMX \rightarrow HMY by the formula HMF (\alpha ) = f \circ \alpha . In general, HMX is not compact.
Let us fix some n \in \BbbN . For every compactum Z consider
HMn(Z) =
\Bigl\{
f \in HM(Z) | there exist 0 = t1 < . . . < tn+1 = 1
with f | [ti, ti+1) \equiv zi \in Z, i = 1, . . . , n
\Bigr\}
.
The constructions HMn define normal functors in \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} [10] (2.5.2).
Zarichnyi has asked if there exists a normal functor in \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} which contains all functors HMn
as subfunctors (see [10]). Such a functor H was constructed in [7]. The main aim of this section is to
show that the functor H does not admit a tensor product. Let us remind the construction of H from
[7]. Let X be a compactum. By C(X) we denote the Banach space of all continuous functions \varphi :
X \rightarrow \BbbR with the usual \mathrm{s}\mathrm{u}\mathrm{p}-norm: \| \varphi \| = \mathrm{s}\mathrm{u}\mathrm{p}\{ | \varphi (x)| | x \in X\} . We denote the segment [0, 1] by I.
For a compactum X let us define the uniformity of HMX. For each \varphi \in C(X) and a, b \in [0, 1]
with a < b we define a function \varphi (a,b) : HMX \rightarrow \BbbR by the following formula:
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
856 T. RADUL
\varphi (a,b) =
1
b - a
b\int
a
\varphi \circ \alpha (t)dt for some \alpha \in HMX.
Define
SHM (X) = \{ \varphi (a,b) | \varphi \in C(X) and (a, b) \subset (0, 1)\} .
For \varphi 1, . . . , \varphi n \in SHM (X) define a pseudometric \rho \varphi 1,...,\varphi n on HMX by the formula
\rho \varphi 1,...,\varphi n(f, g) = \mathrm{m}\mathrm{a}\mathrm{x}\{ | \varphi i(f) - \varphi i(g)| | i \in \{ 1, . . . , n\} \} ,
where f, g \in HMX. The family of pseudometrics
\scrP = \{ \rho \varphi 1,...,\varphi n | n \in \BbbN , where \varphi 1, . . . , \varphi n \in SHM (X)\} ,
defines a totally bounded uniformity \scrU \scrH \scrM \scrX of HMX (see [7]).
For each compactum X we consider the uniform space (HX,\scrU HX) which is the completion of
(HMX,\scrU \scrH \scrM \scrX ) and the topological space HX with the topology induced by the uniformity \scrU HX .
Since \scrU HMX is totally bounded, the space HX is compact.
Let f : X \rightarrow Y be a continuous map. Define a map HMf : HMX \rightarrow HMY by the formula
HMf(\alpha ) = f \circ \alpha for all \alpha \in HMX. It was shown in [7] that the map HMf : (HMX,\scrU HMX) \rightarrow
\rightarrow (HMY,\scrU HMY ) is uniformly continuous. Hence there exists a continuous map Hf : HX \rightarrow HY
such that Hf | HMX = HMf. It is easy to see that H : \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} \rightarrow \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} is a covariant functor and
HMn is a subfunctor of H for each n \in \BbbN .
Let us remark that the family of functions SHM (X) embeds HMX in the product of closed
intervals
\prod
\varphi (a,b)\in SHM (X)
I\varphi (a,b)
, where I\varphi (a,b)
= [\mathrm{m}\mathrm{i}\mathrm{n}x\in X | \varphi (x)| ,\mathrm{m}\mathrm{a}\mathrm{x}x\in X | \varphi (x)| ]. Thus, the space
HX is the closure of the image of HMX. We denote by p\varphi (a,b)
: HX \rightarrow I\varphi (a,b)
the restriction
of the natural projection. Let us remark that the function Hf could be defined by the condition
p\varphi (a,b)
\circ Hf = p(\varphi \circ f)(a,b) for each \varphi (a,b) \in SHM (Y ).
We will use certain properties of the functor H proved in [7]. Since the functor H preserves
embeddings, we can identify the space FA with Fi(FA) \subset FX for each closed subset A \subset X,
where i : A \rightarrow X is the natural embedding. We can define for each \alpha \in HX the closed set
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\alpha = \cap \{ A is a closed subset of X such that \alpha \in HA\} . Since H preserves intersection, we have
\alpha \in H(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\alpha ).
By D we denote two-point space \{ 0, 1\} with discrete topology. For n \in \BbbN define \alpha n0 and
\alpha n1 \in HMD \subset HD as follows: \alpha n0 (s) = 0, \alpha n1 (s) = 1, if s \in
\biggl[
2i
2n
,
2i+ 1
2n
\biggr)
and \alpha n0 (s) = 1;
\alpha n1 (s) = 0, if s \in
\biggl[
2i+ 1
2n
,
i+ 1
n
\biggr)
, where i \in \{ 0, . . . , n - 1\} , s \in [0, 1). For k, j \in \{ 0, 1\} and
n \in \BbbN define \alpha nk,j \in HM(D \times D) \subset H(D \times D) by the formula \alpha nk,j(s) = (\alpha nk(s), \alpha
n
j (s)). Denote
\varrho l = H(\mathrm{p}\mathrm{r}l) : H(D \times D) \rightarrow HD, where l \in \{ 1, 2\} and \mathrm{p}\mathrm{r}l : D \times D \rightarrow D are natural projections.
Obviously, we have \varrho 1(\alpha nk,j) = \alpha nk and \varrho 2(\alpha nk,j) = \alpha nj .
Lemma 1.1. We have (\varrho 1)
- 1(\alpha nk) \cap (\varrho 2)
- 1(\alpha nj ) = \{ \alpha nk,j\} .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
MONADS AND TENSOR PRODUCTS 857
Proof. We will prove the lemma for k = j = 0. The proof is the same for other cases. Consider
any \gamma \in (\varrho 1)
- 1(\alpha n0 ) \cap (\varrho 2)
- 1(\alpha n0 ). Firstly, let us show that \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} \gamma \subset \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\alpha n0,0) = \{ (0; 0), (1; 1)\} .
Suppose the contrary. We can assume that (0; 1) \in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} \gamma . Consider a function \psi : D \times D \rightarrow \BbbR
such that \psi (0; 1) = 1 and \psi (k; l) = 0 for each (k; l) \not = (0; 1). By Lemma 2.5 from [6] there
exists a > 0 such that p\psi (0,1)
(\gamma ) \geq a. For r \in \{ 0, . . . , n - 1\} define functions \varphi 2r, \varphi 2r+1 :
D \rightarrow \BbbR as follows: \varphi 2r(0) = 1, \varphi 2r(1) = 0 and \varphi 2r+1(0) = 0, \varphi 2r+1(1) = 1. For k \in \{ 1, 2\}
and l \in \{ 0, . . . , 2n - 1\} we consider the functions \varphi kl = \varphi l \circ \mathrm{p}\mathrm{r}k : D \times D \rightarrow \BbbR . Choose a
neighborhood V of \gamma defined as follows: V = \{ \gamma \prime \in H(D\times D) | | p\psi (0,1)
(\gamma ) - p\psi (0,1)
(\gamma \prime )| < a
2
and\bigm| \bigm| \bigm| \bigm| \bigm| p\varphi k
r( r
2n , r+1
2n )
(\gamma ) - p\varphi k
r( r
2n , r+1
2n )
(\gamma \prime )
\bigm| \bigm| \bigm| \bigm| \bigm| < a
4n
for each k \in \{ 1, 2\} and r \in \{ 0, . . . , 2n - 1\} .
Consider any \gamma 1 \in HM(D \times D) \cap V. Since | p\psi (0,1)
(\gamma ) - p\psi (0,1)
(\gamma 1)| <
a
2
, we have m\{ t \in
\in [0, 1) | \gamma 1(t) = (0; 1)\} >
a
2
. Hence there exists r \in \{ 0, . . . , 2n - 1\} such that
m
\Biggl\{
t \in
\biggl(
r
2n
,
r + 1
2n
\biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \gamma 1(t) = (0; 1)
\Biggr\}
>
a
4n
.
If r = 2l for some l \in \{ 0, . . . , n - 1\} we have p\varphi 2
r( r
2n , r+1
2n )
(\gamma 1) = n
\int r+1
2n
r
2n
\varphi 2
r \circ \gamma 1(t)dt < 1 - a
2
.
But p\varphi 2
r( r
2n , r+1
2n )
(\gamma ) = p\varphi r\circ pr
2( r
2n , r+1
2n )
(\gamma ) = p\varphi
r( r
2n , r+1
2n )
\circ \varrho 2(\gamma ) = p\varphi
r( r
2n , r+1
2n )
(\alpha n0 ) = 1 and we
obtain a contradiction with the definition of V.
If r = 2l+1 for some l \in \{ 0, . . . , n - 1\} we obtain a contradiction using the function \varphi 1
r . Hence
we have the inclusion \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} \gamma \subset \{ (0; 0), (1; 1)\} .
Consider any \varphi \in C(D \times D) and (a, b) \subset (0, 1). Define \psi \in C(D) as follows: \psi (i) = \varphi (i; i)
for i \in D and put \xi = \psi \circ \mathrm{p}\mathrm{r}1. Since \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} \gamma \subset \{ (i; i)| i \in D\} , we have p\varphi (a,b)
(\gamma ) = p\xi (a,b)(\gamma )
by Lemma 2.3 from [6]. Then p\varphi (a,b)
(\gamma ) = p\xi (a,b)(\gamma ) = p\psi (a,b)(\alpha
n
0 ) = p\xi (a,b)(\alpha
n
0,0) = p\varphi (a,b)
(\alpha n0,0).
Hence \gamma = \alpha n0,0.
Theorem 1.1. There is no continuous map t : HD \times HD \rightarrow H(D \times D) such that \varrho i \circ t = si
for each i \in \{ 1, 2\} , where si : HD \times HD \rightarrow HD is the natural projection.
Proof. Suppose that there exists such map. It is easy to check that both sequences (\alpha n0 ) and
(\alpha n1 ) converge to \alpha \in HD defined as follows p\varphi (a,b)
(\alpha ) =
1
2
(\varphi (0) + \varphi (1)) for each \varphi \in C(D)
and (a, b) \subset (0, 1). Since t is continuous, both sequences t(\alpha n0 , \alpha
n
0 ) and t(\alpha n0 , \alpha
n
1 ) converge to
\beta = t(\alpha , \alpha ). We have t(\alpha n0 , \alpha
n
0 ) = \alpha n0,0 and t(\alpha n0 , \alpha
n
1 ) = \alpha n0,1 by Lemma 1.1. Consider a function \varphi :
D \times D \rightarrow \BbbR defined as follows: \varphi (0; 0) = \varphi (1; 1) = 0 and \varphi (1; 0) = \varphi (0; 1) = 1. Then we have
p\varphi (0,1)
(\alpha n0,0) = 0 \not = 1 = p\varphi (0,1)
(\alpha n0,1) for each n \in \BbbN . We obtain a contradiction and the theorem is
proved.
2. Iterated functors. Let F be a functor. By F 2 we denote the second iteration F \circ F of the
functor F. If we have a family \{ X\alpha \} , by p\alpha :
\prod
\beta
X\beta \rightarrow X\alpha , \mathrm{p}\mathrm{r}\alpha :
\prod
\beta
FX\beta \rightarrow FX\alpha and \mathrm{p}\mathrm{r}2\alpha :\prod
\beta
F 2X\beta \rightarrow F 2X\alpha we denote the corresponding natural projections.
Theorem 2.1. Let F be a functor which admits a tensor product. Then F 2 admits a tensor
product.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
858 T. RADUL
Proof. Let \otimes \{ X\alpha \} :
\prod
\alpha
FX\alpha \rightarrow F
\Bigl( \prod
\alpha
X\alpha
\Bigr)
be a tensor product for a family \{ X\alpha \} and a
functor F. Define a map \otimes 2
\{ X\alpha \} :
\prod
\alpha
F 2X\alpha \rightarrow F 2
\Bigl( \prod
\alpha
X\alpha
\Bigr)
by formula \otimes 2
\{ X\alpha \} = F (\otimes \{ X\alpha \} ) \circ
\circ \otimes \{ FX\alpha \} . The naturality of \otimes 2 is obvious.
For each \alpha we have F 2p\alpha \circ \otimes 2
\{ X\alpha \} = F 2p\alpha \circ F (\otimes \{ X\alpha \} )\circ \otimes \{ FX\alpha \} = F (Fp\alpha \circ \otimes \{ X\alpha \} )\circ \otimes \{ FX\alpha \} =
= F (\mathrm{p}\mathrm{r}\alpha ) \circ \otimes \{ FX\alpha \} = \mathrm{p}\mathrm{r}2\alpha . Hence the operation \otimes 2 defines a tensor product for the functor F 2.
Now we consider the hyperspace functor \mathrm{e}\mathrm{x}\mathrm{p} . For a compactum X by \mathrm{e}\mathrm{x}\mathrm{p}X we denote the set
of nonempty compact subsets of X provided with the Vietoris topology. A base of this topology
consists of the sets of the form \langle U1, . . . , Un\rangle =
\Bigl\{
A \in \mathrm{e}\mathrm{x}\mathrm{p} X | A \subset
\bigcup n
i=1
Ui, A \cap Ui \not = \varnothing for each
i \in \{ 1, . . . , n\}
\Bigr\}
, where U1, . . . , Un are open in X. The space \mathrm{e}\mathrm{x}\mathrm{p} X is called the hyperspace of X.
For a continuous mapping f : X \rightarrow Y the mapping \mathrm{e}\mathrm{x}\mathrm{p} f : \mathrm{e}\mathrm{x}\mathrm{p} X \rightarrow \mathrm{e}\mathrm{x}\mathrm{p} Y is defined by the
formula \mathrm{e}\mathrm{x}\mathrm{p} f(A) = fA \in \mathrm{e}\mathrm{x}\mathrm{p} Y, A \in \mathrm{e}\mathrm{x}\mathrm{p} X. It is easy to see that this defines a functor \mathrm{e}\mathrm{x}\mathrm{p} :
\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} \rightarrow \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p} (the hyperspace functor). It is well known that the functor \mathrm{e}\mathrm{x}\mathrm{p} define the natural
transformations s : \mathrm{I}\mathrm{d}Comp \rightarrow \mathrm{e}\mathrm{x}\mathrm{p} and u : \mathrm{e}\mathrm{x}\mathrm{p}2 \rightarrow \mathrm{e}\mathrm{x}\mathrm{p} as follows: sX(x) = \{ x\} for each x \in X,
uX(\scrA ) = \cup \scrA , \scrA \in \mathrm{e}\mathrm{x}\mathrm{p}2 X. Then the triple \BbbH = (\mathrm{e}\mathrm{x}\mathrm{p}, s, u) is a monad (see [10] for more
information). Hence the hyperspace functor admits a tensor product and, by Theorem 2.1, the iterated
functor \mathrm{e}\mathrm{x}\mathrm{p}2 admits tensor product too.
Let us remark that there exists a unique natural transformation \eta : \mathrm{I}\mathrm{d}Comp \rightarrow \mathrm{e}\mathrm{x}\mathrm{p}2 defined as
follows \eta X = s \mathrm{e}\mathrm{x}\mathrm{p}X \circ sX = \mathrm{e}\mathrm{x}\mathrm{p} sX \circ sX for each compactum X. We have \eta X(x) = \{ \{ x\} \} for
x \in X.
Theorem 2.2. There is no natural transformation \mu : \mathrm{e}\mathrm{x}\mathrm{p}4 \rightarrow \mathrm{e}\mathrm{x}\mathrm{p}2 such that \mu \circ \mathrm{e}\mathrm{x}\mathrm{p}2 \eta =
= \mu \circ \eta \mathrm{e}\mathrm{x}\mathrm{p}2 =1exp2 .
Proof. Suppose the contrary. Consider X = \{ a, b, c, d\} and \alpha = \{ \{ \{ \{ a\} , \{ b\} \} , \{ \{ d\} , \{ c\} \} \} \} \in
\in \mathrm{e}\mathrm{x}\mathrm{p}4X. Define maps f1, f2 : X \rightarrow \{ 0, 1\} as follows: f1(a) = f1(b) = 1, f1(c) = f1(d) =
= 0 and f2(a) = f2(c) = 0, f2(b) = f1(d) = 1. Then \mathrm{e}\mathrm{x}\mathrm{p}4 f1(\alpha ) = \{ \{ \{ \{ 0\} \} , \{ \{ 1\} \} \} \} =
= \mathrm{e}\mathrm{x}\mathrm{p}2 \eta \{ 0, 1\} (\{ \{ 0, 1\} \} ) and we have \mu \{ 0, 1\} \circ \mathrm{e}\mathrm{x}\mathrm{p}4 f1(\alpha ) = \{ \{ 0, 1\} \} . Since \mu is a natural
transformation, we obtain that \mathrm{e}\mathrm{x}\mathrm{p}2 f1 \circ \mu X(\alpha ) = \{ \{ 0, 1\} \} . On the other hand \mathrm{e}\mathrm{x}\mathrm{p}4 f2(\alpha ) =
= \{ \{ \{ \{ 0\} , \{ 1\} \} \} \} = \eta \mathrm{e}\mathrm{x}\mathrm{p}2\{ 0, 1\} (\{ \{ 0\} , \{ 1\} \} ) and we have that \mathrm{e}\mathrm{x}\mathrm{p}2 f2 \circ \mu X(\alpha ) = \{ \{ 0\} , \{ 1\} \} . It
is easy to check that \mu X(\alpha ) = \{ \{ a, c\} , \{ b, d\} \} .
But if we consider maps g1, g2 : X \rightarrow \{ 0, 1\} defined by equalities g1 = f1 and g2(a) = g2(d) =
= 0, g2(b) = g2(c) = 1, we obtain that \mu X(\alpha ) = \{ \{ a, d\} , \{ b, c\} \} using the same arguments as
before. Hence we have a contradiction.
Corollary 2.1. The functor \mathrm{e}\mathrm{x}\mathrm{p}2 can not be completed to a monad.
Let us remark that the functors \mathrm{e}\mathrm{x}\mathrm{p} and \mathrm{e}\mathrm{x}\mathrm{p}2 are normal (see, for example, [10]).
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ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
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Received 14.07.15,
after revision — 05.02.16
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| id | umjimathkievua-article-1740 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:11:44Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/58/f3d41d205eb4022c1308982bf1e1b658.pdf |
| spelling | umjimathkievua-article-17402019-12-05T09:25:15Z Monads and tensor products Монади та тензорнi добутки Radul, T. N. Радул, Т. Н. M. Zarichnyi defined an operation of tensor product for each functor that can be complemented to a monad. We investigate the existence of tensor product for functors which cannot be complemented to monads. М. Зарiчний означив операцiю тензорного добутку для кожного функтора, що доповнюється до монади. У цiй статтi дослiджено iснування тензорного добутку для функторiв, якi не можна доповнити до монади. Institute of Mathematics, NAS of Ukraine 2017-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1740 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 6 (2017); 854-859 Український математичний журнал; Том 69 № 6 (2017); 854-859 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1740/722 Copyright (c) 2017 Radul T. N. |
| spellingShingle | Radul, T. N. Радул, Т. Н. Monads and tensor products |
| title | Monads and tensor products |
| title_alt | Монади та тензорнi добутки |
| title_full | Monads and tensor products |
| title_fullStr | Monads and tensor products |
| title_full_unstemmed | Monads and tensor products |
| title_short | Monads and tensor products |
| title_sort | monads and tensor products |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1740 |
| work_keys_str_mv | AT radultn monadsandtensorproducts AT radultn monadsandtensorproducts AT radultn monaditatenzornidobutki AT radultn monaditatenzornidobutki |