Questions tagged [adjoint-functors]
For questions about adjoint functors from category theory. Use in conjunction with the tag (category-theory).
803 questions
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Connection between Cayley's theorem and proposition that every group is quotient of a free group
So Cayley's theorem tells us that every group $G$ is a subgroup of $Aut_{Set}(U(G))$, where $U: Grp \rightarrow Set$ is forgetful functor, mapping every group to its underlying set, and every group ...
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When are the split morphisms given by the triangle identities isomorphisms?
Given an adjunction $(L,R, \eta, \epsilon)$ between categories $C$, $D$, the triangle identities tell us that the natural transformations $\epsilon L\colon LRL \to L$, $L \eta \colon L \to LRL$, $R\...
- 525
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Is there an adjunction of the form $[\mathcal{C}^{\text{op}}, \text{Set}^{\text{op}}] \leftrightarrows [\mathcal{C}^{\text{op}}, \text{Set}]$?
Let $\mathcal{C}$ be a category.
Consider the following four functors:
$$
\newcommand\op[1]{{#1}^{\operatorname{op}}}
\newcommand\Set{\operatorname{Set}}
\newcommand\Hom{\operatorname{Hom}}
\...
- 4,970
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1
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Do both sides of a quillen equivalence preserve homotopy colimits?
Suppose I have $L: M_1\leftrightarrow M_2:R$ a Quillen equivalence between model categories (if this helps I am happy to assume these are combinatorial). It is direct (by construction one might say) ...
- 1,787
1
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Proving the adjointness of an inclusion functor
I recently read a post (either on here or MathOverflow) that mentioned that the fraction field functor $$\mathrm{Frac:Int \to Fld}, R\mapsto R^2/\{(a,a),a\in R\}$$(where $\mathrm{Int}$ is the category ...
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Fibred Products
I am learning fibred categories and in B. Jacobs's book, Categorical logic and Type Theory, Definition 1.9.4 defines fibred products or products inside a fibration $P \colon E \to B$ by existence of a ...
- 141
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Properties of left and right adjoint functors to pushforward functor from a divisor
Let $X$ be a Noetherian variety, and $D$ a Cartier divisor. Let $i:D\hookrightarrow X$ be the inclusion. Let $i_* : Coh(D)\to Coh(X)$ be the functor between derived category of bounded coherent ...
0
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Sufficient conditions *against* the existence of a cohesive structure on a presheaf topos
I recently came across the notion of a cohesive topos: a topos $\mathcal{X}$ is cohesive over Set if there is a string of adjunctions
$$
\Pi_0 \dashv D \dashv \Gamma \dashv C \colon \mathcal{X} \to \...
- 849
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Equivalent formulation of naturality in a two-variable adjunction. Is it mentioned somewhere in the literature?
$\def\A{\mathsf{A}}
\def\B{\mathsf{B}}
\def\C{\mathsf{C}}
\def\D{\mathsf{D}}
\def\op{\mathrm{op}}$
Here is an equivalent way of formulating the naturality condition of an adjunction:
[R, Lemma 4.1.3]....
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1
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Hom sets, isomorphisms and equivalence of categories [duplicate]
Let $\mathcal{C}$ and $\mathcal{D}$ be two categories and let $F:\mathcal{C} \to \mathcal{D}$ and $G:\mathcal{D} \to \mathcal{C}$ be a pair of functors that implement an equivalence of categories. It ...
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74
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Natural retractions of the free monoid functor
Let $\mathcal{C}$ be a cocomplete symmetric-monoidal closed category. Write
$$T\colon\mathcal{C}\to\mathcal{C}, \qquad T(X)=\coprod_{n\geq 0} X^{\otimes n}$$
for the free monoid monad, with unit $i_X\...
- 121
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Weibel Exercise 8.4.3 $KC$ left adjoint to the forgetful functor $U:\mathcal{SA}\to \mathcal A^{\Delta_s^{op}}$
Given a semi-simplicial object $B$ in an abelian category $\mathcal A$, we have associated complex $C(B)$ and simplicial object $KC(B)$. Show that $KC$ is left adjoint to the forgetful functor from ...
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morphism between resolutions for a monad
Let (T, η, µ) be a monad over a category C. A resolution for (T, η, µ) is a
category D and an adjunction (F, G, η, ε) from C to D such that T = GF and µ = GεF. According to Definition 5.4.2 of this ...
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Thinking of the diagonal function $\text{Set} \to \text{Set} \times \text{Set}$ as a forgetful functor?
I'm learning category theory and getting used to finding free-forgetful analogies to adjunction between functors. I've done an exercise tasking me to find left and right adjoints to the diagonal ...
2
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Unit and counit of an adjunction coming from an equivalence?
I'm reading a book on category theory by Tom Leinster and I've encountered something a little weird. It states that when $(F, G, \eta, \epsilon)$ is an equivalence, it is true that $F$ is left adjoint ...