I must prove that a functor $G:Y\to X$ that preserves limits has a left adjoint $F$, where $X$ and $Y$ are preorder categories with all limits (i.e. all products). I don't see how to define $F$ with this data; I thought of sending every $x\in X$ to the initial or terminal object of $Y$ but it doesn't work, and I can't see an other choice. Can anyone give me an hint?
2026-03-28 07:36:30.1774683390
Exercise about adjunction of preorders
79 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in CATEGORY-THEORY
- (From Awodey)$\sf C \cong D$ be equivalent categories then $\sf C$ has binary products if and only if $\sf D$ does.
- Continuous functor for a Grothendieck topology
- Showing that initial object is also terminal in preadditive category
- Is $ X \to \mathrm{CH}^i (X) $ covariant or contravariant?
- What concept does a natural transformation between two functors between two monoids viewed as categories correspond to?
- Please explain Mac Lane notation on page 48
- How do you prove that category of representations of $G_m$ is equivalent to the category of finite dimensional graded vector spaces?
- Terminal object for Prin(X,G) (principal $G$-bundles)
- Show that a functor which preserves colimits has a right adjoint
- Show that a certain functor preserves colimits and finite limits by verifying it on the stalks of sheaves
Related Questions in LIMITS-COLIMITS
- Show that a certain functor preserves colimits and finite limits by verifying it on the stalks of sheaves
- Prove that a "tensor product" principal $G$-bundle coincides with a "pullback" via topos morphism
- What prevents me from starting the cone in the objects mapped by the diagram
- Two morphisms $f, g : M \to L$ are equal as long as they are equal under the limit $L$.
- Separating coproducts by its index set
- Preservation of coproducts by coproducts
- Categorification of commutative sum
- Colimit of a directed system of modules
- Pullback square with two identical sides
- Subring of an inverse limit is the entire thing if maps to pieces are surjective
Related Questions in ADJOINT-FUNCTORS
- Show that a functor which preserves colimits has a right adjoint
- How do I apply the Yoneda lemma to this functor?
- Determining Left Adjoint of Forgetful Functor from $\tau_{*}$ to $\tau$
- What is the left adjoint to forgetful functor from Mod(R) to Ab
- Does the left adjoint to the forgetful functor have another left adjoint?
- Is coreflectiveness transitive?
- Group algebra functor preserves colimits
- Intuition for remembering adjunction chirality
- Does the inverse image sheaf functor has a left adjoint?
- Significance of adjoint relationship with Ext instead of Hom
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?