Redirected from "cotensors".
Contents
This entry is about the formal dual to tensoring in the generality of category theory. For the different concept of cotensor product of comodules see there.
Context
Enriched category theory
Limits and colimits
limits and colimits
1-Categorical
-
limit and colimit
-
limits and colimits by example
-
commutativity of limits and colimits
-
small limit
-
filtered colimit
-
sifted colimit
-
connected limit, wide pullback
-
preserved limit, reflected limit, created limit
-
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
-
finite limit
-
Kan extension
-
weighted limit
-
end and coend
-
fibered limit
2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Idea
In a closed symmetric monoidal category the internal hom satisfies the natural isomorphism
for all objects (prop.). If we regard as a -enriched category we write and this reads
If we now pass more generally to any -enriched category then we still have the enriched hom object functor . One says that is powered over if it is in addition equipped also with a mixed operation such that behaves as if it were a hom of the object into the object in that it comes with natural isomorphisms of the form
Definition
Definition
Let be a closed monoidal category. In a -enriched category , the power of an object by an object is an object with a natural isomorphism
where is the -valued hom of and is the internal hom of .
We say that is powered or cotensored over if all such power objects exist.
Properties
- Powers are a special sort of weighted limit: in particular, where the domain is the unit -category. Conversely, all weighted limits can be constructed from powers together with conical limits. The dual colimit notion of a power is a copower.
Examples
In 1-category theory
-
itself is always powered over itself, with .
-
Every locally small category ( ) with all products is powered over Set: the powering operation
of an object by a set forms the -fold cartesian product of with itself, where is the cardinality of .
The defining natural isomorphism
is effectively the definition of the product (see limit).
-
In a 2-category (seen as a -enriched category), powers by the walking arrow are ways to internalize ‘generalized arrows’ of a given object . Specifically, , called the object of arrows of is, when it exists, an object such that:
Thus generalized elements of correspond to 2-cells between generalized elements of , explaining why can be considered a ‘view from the inside’ of the internal structure of .
Powering of -toposes over -groupoids
We discuss how the powering of -toposes over is given by forming mapping stacks out of locally constant -stacks. All of the following formulas and their proofs hold verbatim also for Grothendieck toposes, as they just use general abstract properties.
Let be an -topos
-
with terminal geometric morphism denoted
(1)
where the inverse image constructs locally constant -stacks,
-
and with its internal hom (mapping stack) adjunction denoted
(2)
for .
Notice that this construction is also -functorial in the first argument: is the morphism which under the -Yoneda lemma over (which is large but locally small, so that the lemma does apply) corresponds to
By definition, for any and the powering] is the (∞,1)-limit over the diagram constant on
while the tensoring is the (∞,1)-colimit over the diagram constant on
Proposition
The powering of over is given by the mapping stack out of the locally constant -stacks:
in that this operation has the following properties:
-
For all and we have a natural equivalence
-
In its first argument the operation
-
sends the terminal object (the point) to the identity:
(3)
-
sends -colimits to -limits:
(4)
where all equivalences shown are natural.
Proof
For the first statement to be proven, consider the following sequence of natural equivalences:
For the second statement, recall that hom-functors preserve limits in that there are natural equivalences of the form
(5)
and that -toposes have universal colimits, in particular that the product operation is a left adjoint (2) and hence preserves colimits:
(6)
With this, we get the following sequences of natural equivalences:
This implies (4) by the -Yoneda lemma over (which is large but locally small, so that the lemma does apply).
Finally (3) is immediate from the fact that preserves the terminal object, by definition:
References
Textbook accounts: