Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

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Contruction of Weighted Colimit in a 2-category

On page 306 of Kelly's Elementary Observations on 2-categorical Limits, it is explained that a weighted limit $\{F, G\}$ in a 2-category can be constructed as the equalizer of $v$ and $w$ in $$ (3.2) \...
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Naturality as functoriality on arrow category

Here it is said that a natural transformation $\varphi:F\Rightarrow G$ is the same as a function $\varphi_0:\mathrm{Ob}\mathcal A\rightarrow \mathrm{Mor}\mathcal B$ satisfying ${\rm dom}\circ\varphi=F,...
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31 views

Adjunction with reversed elements

There is an adjunction between $L$ and $R$ when: $$ \text{Hom}(LA,B) \approx \text{Hom}(A,RB) $$ Is there something related we can say when instead we have: $$ \text{Hom}(B,LA) \approx \text{Hom}(A,...
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What are the group objects in the category of finite sets and bijections, and its functor category?

An object $G$ in a category $\mathcal{C}$ is called a group object if, given any object $X$ in $\mathcal{C}$, there is a group structure on the morphisms $\operatorname{hom}\left(X,G\right)$ such that ...
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35 views

A linear category is a Vect-module

I would like to know how to show that any linear category is a $\mathrm{Vec}$-module. Here $\mathrm{Vect}$ denotes a category of finite dimensional vector spaces. More general statement can be found ...
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Existence of adjoints with commutativity condition

Are there (non-trivial) examples of adjunctions $F \operatorname{\dashv} G$ with unit and counit $\eta$, $\epsilon$ such that $$\eta GF = GF \eta$$ and $$FG \epsilon = \epsilon FG,$$ and if so, what ...
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63 views

Deducing the existence of particular functions $\mathbb{N}\longrightarrow\mathbb{Q}$ in the context of Tom Leinster's “Rethinking Set Theory”

This question concerns the set theory given by Tom Leinster in his paper "Rethinking Set Theory," available here: http://arxiv.org/abs/1212.6543 In this paper, axioms for set theory are given in the ...
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Orthogonal Factorization Systems and functoriality

In definition 1.1 of these notes on factorization systems, (III) calls the factorization functorial if given the solid diagram described, there's a unique horizontal arrow making both squares commute. ...
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67 views

What limits/colimits are preserved by the Yoneda embedding?

I know that the contravariant Yoneda embedding $X\mapsto \mathcal{C}(-,X)$ preserves all small limits that exist in $\mathcal{C}$. I guess it follows that the covariant Yoneda embedding preserves all ...
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56 views

Associativity of the tensor product of bimodules

Let $A_0,\dots,A_n$ be algebras over some fixed commutative ring $k$ (you may assume $k=\mathbb{Z}$ for simplicity). Let $M_i$ be an $(A_{i-1},A_i)$-bimodule for $i=1,\dots,n$. A multilinear map from $...
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An asymmetry in the Galois connection between topologies and sequential convergences?

On a set $X$ consider a relation $c \subseteq X^{\mathbb{N}} \times X$ and for $((x_n), x) \in c$ write $x_n \to_c$ x. Such a relation $c$ is a sequential convergence if (i) $x \to_c x$ for all $x \in ...
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Simplicial homotopy's exponential law

The following was cited from Simplicial Homotopy Theory by John F. Jardine and Paul Gregory Goerss. We have an adjunction $$\hom_{\mathbf{S}}(X\times Y,Z)\simeq \hom_{\mathbf{S}}(Y,\mathbf{Hom}(X,Z))$$...
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Functorial construction of the convolution algebra of measures on a group

Let $G$ be a lcoally compact group and $C_c(G) = \lim_{K \subset G} C(K)$ its space of continuous functions with compact support endowed with the topology of the limit of banach spaces $C(K)$ with $K$ ...
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35 views

Action groupoid as $G\rightrightarrows \textrm{Bij}(X)$?

Let $G$ be a group and $X$ a set. A left action of $G$ on $X$ can be thought either as a map $G\times X\longrightarrow X$, $(g, x)\longmapsto g\cdot x$, satisfying: $(i)$ $g\cdot (h\cdot x)=(gh)\cdot ...
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Epimorphisms and faithful functors in a rigid abelian tensor category

Let $\mathsf{C}$ be a rigid abelian tensor category in the sense of Deligne and Milne's notes (p.9). Let $\mathbf{1}$ denote the identity object in $\mathsf{C}$ with respect to $\otimes$. The abelian ...
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Subcategory of category of Module satisfies SSA?

Let $\mathcal{D}=Mod-A$ be a category of module over an Algebra $A$. and $\mathcal{C}$ be a subcategory of $\mathcal{D}$, I have the following questions:- Is $C$ complete category? Is $\mathcal{C}$ ...
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Composing functors with natural transformations

So I'm doing a project in Category Theory. I fully understand natural transformations and functors, but what does it mean to compose them, for example in the monad axioms where you have something like ...
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Question Janelidze and Tholen's 'Beyond Barr Exactness: Effective Descent Morphisms'

I have some questions about Janelidze and Tholen's paper Beyond Barr Exactness: Effective Descent Morphisms. First of all, they remark at the end of the introduction: Throughout this chapter $\...
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1answer
41 views

Polynomial functors preserving ω-colimits

I'm working my way through Steve Awodey's Category Theory book and on p.271, Proposition 10.12 says: If the category $S$ has an initial object $0$ and colimits of diagrams of type $\omega$ (call them ...
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71 views

Is there a category in which, between any two objects, there is a unique morphism?

I am interested in knowing information about such a category (if it is well-defined). Does there exist a category $\mathcal{C}$, in which there is a unique morphism between any two objects in it?
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adjoint functor and subcategories

First at all i have to say I am very new to categories (just basic definition). My Question:- We have two categories ( $\mathcal{C},\mathcal{R}$) object in both of these two categories are module ( ...
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33 views

Extension Operator.

I am working on my thesis about completion and extensions from an algebraic point of view. We have the closure operator which takes subsets to subsets with 3 criterias to meet $X\subseteq C(X)$ $X\...
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Presheaf and copresheaf categories on finite sets

Briefly: do they agree? In more detail: denote by $\mathbf{Finset}$ the category of finite sets, and by $\mathbf{Set}$ the category of sets. I want to know what the functor categories $[\mathbf{...
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63 views

Non-monadic adjunction

Could someone give some examples of a non-monadic adjunctions please? Possibly explaining why they are not monadic and how they contradict the monadicity theorem? Thanks!
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Establish canonical isomorphism in Set category

Using notation $A^B := \mathsf{Mor}_{\mathcal{SET}}(B, A)$ establish canonical isomorphisms for any sets $X, Y$ and $Z$: $$ (Z^Y)^X \cong Z^{Y \times X} \; , \;(Z \times Y)^X \cong Z^X \times Y^X.$$ ...
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71 views

Formal Proof that $f^{-1} \circ f = id_x \ , \forall f$

Given $f$ as an invertible function with domain $X$ and codomain $Y$, then we can say $$f^{-1}(f(x)) = x $$ Or since they are both logically equivalent $$ f(f^{-1}(x)) = x $$ This can also be ...
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Adjointness of internal contravariant Hom in symmetric monoidal categories.

Let consider a closed symmetric monoidal cateogry, $\mathscr C,\otimes$, with adjunction $(X\otimes-)\dashv\mathrm{Hom}(X,-)$ for all objects $X$. The following isomorphis, valid in categories such ...
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Functor between Category and its Free Strict Monoidal Category

Let $C$ be a category and let $\sum(C)$ denote the free strict monoidal category over $C$. According to Wikipedia, the operation $\sum: C\rightarrow\sum(C)$ extends to a 2-monad on $Cat$. Can someone ...
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60 views

Terminal object in the category of sheaves?

Let $\mathsf{C}$ denote one of the categories $\mathsf{Set}$, $\mathsf{Group}$, $\mathsf{Ring}$, or $\mathsf{Mod}_R$, or some other sensible category in which we would like sheaves to take their ...
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split object as coproduct

Let $X$ be an object and $f : Y \to X$ a monomorphism in some category with coproducts. Is there a categorical characterization for objects $Z$ such that $Y + Z \cong X$ and left injection composed ...
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1answer
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If monoid satisifes universal mapping property over $X$, then $X$ generates the monoid

A monoid $M$ satisfies the universal mapping property (UMP) over $X$, if $X \subseteq M$ and for every map $\varphi : X \to N$, where $N$ is another monoid, there exists a unique homomorphism $\varphi ...
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Question about limit of cosimplicial diagram associated with a sheaf

Let $F$ be a presheaf of sets on the usual topological site. Let $\left\{U_i\rightarrow U\right\}$ be covering of $U$. On the second page of Hypercovers and Simplicial Presheaves the authors write the ...
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How to adjoin a terminal object to a category such that each constant morphism factors through it

Let $\mathbf{C}$ be any category. Is there a way to adjoin a terminal object $\ast$ to $\mathbf{C}$ such that each constant morphism factors through any terminal object, i.e., for each constant $f:A\...
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Rings that cannot be representations rings

Given a monoidal category $\mathcal{C}$ one can define the Green ring (or representation ring) $r(\mathcal{C})$ as the abelian group generated by the isomorphism classes $[V]$ of $\mathcal{C}$ modulo ...
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If two categories are equivalent and the one is monoidal, is the other monoidal too?

Pretty much what I ask in the title. Let $\mathcal{C},\mathcal{D}$ be two categories and suppose there exists a fully faithful, surjective-on-objects functor $F:\mathcal{C}\to\mathcal{D}$ (so that $\...
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When are sheafification and the embedding of sheaves into presheaves exact functors?

Let $\text{PreSh}(X, \mathsf{A})$ and $\text{Sh}(X, \mathsf{A})$ be the categories of $\mathsf{A}$-valued presheaves and sheaves on $X$, respectively. Here, $\mathsf{A}$ is a category such that we ...
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A monoid is universal (or free) over its generators iff no nontrivial relations hold among its generators

Let $X$ be any set. A monoid $M$ is called universal over $X$ iff $X \subseteq M$ and for every other monoid $N$ and function $\varphi : X \to N$ there exists a unique extension $\varphi : M \to N$ of ...
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Do $G$-spaces with equivalent orbit categories also have equivalent fundamental categories?

I have heard it mentioned before that $G$-spaces which have equivalent orbit categories must then have equivalent fundamental categories (sometimes called the equivariant fundamental groupoid). This ...
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Is the category of Stone spaces monoidal?

Is there a monoidal structure on the category of compact Hausdorff totally disconnected topological spaces (i.e. Stone spaces)?
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How are image and pre-image different from range and domain respectively?

How are image and pre-image different from range and domain respectively, in Layman's terms (as simple as possible)? Are they basically just keywords that often indicate more nuanced subsets of the ...
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Why is the functor of tensor coalgebra right adjoint to the forgetful functor?

Let $\mathbf{NCoalg}$ be the category of non-unital coassociative conilpotent coalgebras (where "conilpotent" means that for any element there exists a power of comultiplication that vanishes on this ...
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94 views

Construction of 2-limits in 2-categories

Limits in a category can be built as a combination of the basic limits that are products and equalizers. Is there a similar construction for 2-limits in a 2-category? If yes, is it from products and ...
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question about a theorem in Maclane-Moerdijk's “Sheaves in Geometry and Logic”

Like this questioner I am trying to understand the proof of Theorem 2 of Section 5, Chapter I, of MacLane-Moerdijk's "Sheaves in Geometry and Logic". I am wondering, how do you define the functor $L$...
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Left and right endomorphisms in monoidal categories

Let consider a monoidal category $\mathscr C, \otimes$ with unit $I$ and unitors $\lambda_X:I\otimes X\to X$ and $\varrho_X:X\otimes I\to X$. Each endomorphism of the unit $a:I\to I$ induces, on each ...
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Commutations of pullbacks and coproducts?

While unraveling the formalism of descent in basic settings (see this question), I came across two different kinds of isomorphisms which involve pullbacks and coproducts. One is $$X\times_U\...
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On the anti-equivalence of affine schemes with commutative rings

There is an equivalence $\mathbf{Aff}\simeq \mathbf{CRing}^{\text{op}}$ between the category of affine schemes and the category opposite to the category of all commutative rings. If we instead ...
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Rings as internal objects?

Why rings can be seen as monoid objects internal to the category of abelian groups? In the traditional way rings are abelian groups, but I do not understand the role of monoids! I know what internal ...
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1answer
61 views

Are the collections {rings}, {smooth manifolds} larger or smaller than {ab. groups}, {top. manifolds}?

Intuitively, the collection of smooth manifolds feels smaller than that of topological manifolds: they are not just locally nice continuous objects, but even smooth. Similarly, when one finds that an ...
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Ambiguity in the definition of group objects

Given a category $C$ with finite products and a terminal object, the usual definition of a group object $G$ in $C$ specifies certain morphisms along with commutativity of certain diagrams involving ...
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When does this converse of Vopěnka's principle hold?

The $n$Lab page on coreflective subcategories cites a theorem of Adamek and Rosický showing that every colimit-closed full subcategory of a locally presentable category is coreflective. My question is,...