Mathematics

Questions tagged [kan-extensions]

For questions about left, right and absolute Kan extensions from category theory. Use in conjunction with the tag (category-theory).

Kan extensions are another fundamental universal construction in category theory which is closely related to (co)limits, (co)ends and also adjunctions. The slogan "All Concepts are Kan Extensions" explains its ultimate generality in category theory. The idea is to generalize the "notion of extending a function defined on a subset to a function defined on the whole set" into categories and functors. As it sounds possible extensions are not unique and not always exists due to obstructions of various forms. But sometimes there exists one or two "best" approximate extensions called left Kan extension and right Kan extension equip with similar but different universal properties.

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Toy examples for Kan extensions

Background: If $\mathcal{C}$ is a cocomplete category and $f : I \to J$ is a functor between small categories, then $f^* : \mathrm{Hom}(J,\mathcal{C}) \to \mathrm{Hom}(I,\mathcal{C})$ has a left adjoint $\mathrm{Lan}(f)$, the left Kan extension…
Martin Brandenburg
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How to understand the effect of adjoint functors?

I have a good grasp of all different definitions/interpretations of adjoint functors, but still do not know have to interpret the left or right adjoint of a give functor, when it exist. It would be a easy to explain my question through an example.…
Bumblebee
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Description of adjunctions in a 2-category via Kan extensions?

Let $\Psi$ be a strict (for now) 2-category. Let $C,D$ be 0-cells in $\Psi$ ; let $L : C \rightarrow D$ and $R : D \rightarrow C$ be 1-cells; and let $\eta : 1_C \Rightarrow RL$ and $\epsilon : LR \Rightarrow 1_D$ be 2-cells. Claim: Then, if I'm not…
I.A.S. Tambe
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Functoriality for presheaves

Let $\mathcal{C}$ be a small category. Functoriality for presheaves says that for any functor $u\colon\mathcal{C}\to\mathcal{D}$ the precomposition functor $u^*\colon PSh(\mathcal{D})\to PSh(\mathcal {C})$ has two adjoints $u_!, u_*$ on the left and…
JfR_01
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On the two definitions of derived functor in general triangulated category

I'm learning homological algebra using several references books. But I find two definitions of derived functor in general triangulated category. I wonder to know which definition is more generally accepted. Let $\mathcal{D}$ be a triangulated…
Z. He
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What does the Kan extension of a functor along itself look like?

Let $I, J$ be categories and $\varphi : I \to J$ a functor, then we have a pullback functor $\varphi^* : \mathcal{C}^J \to \mathcal{C}^I$ . The left and right Kan extensions along $\varphi$ , denoted by $\operatorname{Lan}$ and $\operatorname{Ran}$…
Long-Ping Li
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Model independent Definition for relative Kan extensions

Kan extensions are really useful in category theory as a means to construct an onslaught of interesting new functors (and adjunctions). For instance, the Nerve-Realization Paradigm can be understood as a left Kan extension along the Yoneda…
h3fr43nd
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Example of Day convolution

By the Eilenberg-Watts theorem there is a monoidal equivalence between the category of $R^e$ -bimodules and the category of cocontinuous, additive endofunctors on the category of $R$ -bimodules. $R$ is a ring and $R^e=R\otimes \overline{R}$ its…
Margaret
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Emily Riehl's proof of density theorem for non-small categories

$\def\C{\mathsf{C}} \def\J{\mathsf{J}} \def\D{\mathsf{D}}$ I am interested in the proof of the density theorem in Riehl's Category Theory in Context [R, 6.5.7]. Before stating the result, I will copy here two results that are invoked in the proof of…
Elías Guisado Villalgordo
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Usage of Yoneda lemma to show ${\rm Lan}_K \dashv \mathcal{E}^K$

I am reading a proof showing that for $K:\mathcal{C}\rightarrow \mathcal{D}$ and $\mathcal{E}$ a category, the functor ${\rm Lan}_K$ is left adjoint to $\mathcal{E}^K$ and there is a thing I don't understand. I was able to do it by showing the…
Julia
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Adjoint Functor Theorem via Kan extensions?

Adjoint functor theorem involves solution set condition and somehow sneaks in some category of elements or comma categories. I hate this way of doing things. Is there any alternate way of doing it, say via Kan extension? Adjoints can be expressed as…
Viśvāmitra
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Does the Yoneda extension of a "representably flat" functor preserve finite limits?

Let me first recap the definitions of flatness, using the nLab's terminology: A functor into $F : \mathcal{C} \to \mathbf{Sets}$ is Set-valued flat when the category of elements $\int E$ is filtered. This implies, and is implied by, left exactness…
Amélia Liao
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Comma categories and Kan Extensions

I believe a copresheaf $W:\mathbf{J} \to \mathsf{Set}$ is the left Kan extension of $\Delta \{ *\}:\operatorname{Elts}W \to \mathsf{Set}$ along the projection $\operatorname{Elts}W \xrightarrow{\pi} \mathbf{J}$. Also $\operatorname{Elts}W$ happens…
Eric Auld
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Left Kan extension of a (co)presheaf along a fully-faithful functor

Let $p: C \to D$ be a fully-faithful functor, let $F: D \to \mathbf{Set}$ be any functor. Is it true that the left Kan extension $\operatorname{Lan}_p Fp$ is given by $$ (\operatorname{Lan}_p Fp)(X) = \{ Ff(y) \in F(X) \mid y \in Fp(Y) \text{ and }…
duck
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Homotopy colimit formula deduced from Coend Quillen bifunctor

Reading the article on Quillen bifunctors on the Nlab ( https://ncatlab.org/nlab/show/Quillen+bifunctor ) I stumbled upon the following claim: Let $\mathcal{C}$ be a combinatorial simplicial model category and let $\mathcal{D}$ be a small category.…
h3fr43nd
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