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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How to construct a G-extension of a category C?

Note: I'm a physicist so this will be phrased somewhat in physics language. Suppose we have a unitary modular tensor category $\mathcal{C}$. In physics language, we can think of $\mathcal{C}$ as ...
3
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0answers
33 views

Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it?

I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
5
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1answer
60 views

Showing epimorphism without using the Freyd-Mitchell Embedding Theorem

In an Abelian category $\mathscr{C}$ consider a commutative diagram as follows: $$\require{AMScd}\begin{CD} 0@>>>\ker f@>\theta>>W @>{f}>> Y\\ @. @. @V{\phi}VV @|{id} \\ @. ...
2
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0answers
34 views

What would be an arrow in category of Hilbert space?

Let $H,K$ be Hilbert spaces. Let $S$ be a Hilbert basis for $H$. (Which means that $S$ is orthogonal and the span of $S$ is dense in $H$.) For each arrow $S\rightarrow K$, there exists a unique ...
4
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83 views

Unique representation of a degenerate simplex

I recently learned about simplicial sets, and now I'm studying some basic properties. I've learned that every degenerate simplex is a degeneracy of a unique non-degenerate (nice) simplex. However, I ...
0
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1answer
45 views

reference request: Category theory

I am sure that a similar question has been asked before, but I make my ideal textbook and situation more specific. I would like a textbook on category theory designed for someone who knows basically ...
1
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1answer
85 views

A categorical first isomorphism theorem

It is known, that for a morphism of universal algebras $f : A \to B$, if $R$ is the congruence relation given by $xRy \Leftrightarrow fx=fy$, then $\operatorname{im} f \cong A/R $. Here is an idea ...
3
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33 views

Is there a name for those concrete categories in which every subset / quotient set inherits the structure of an object in at most one way?

The following situation seems to occur a lot in abstract algebra: We have a category $\mathbf{C}$, concrete over $\mathbf{Set}$, that satisfies: For every object $Y$ of $\mathbf{C}$ and every set ...
3
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64 views
+50

Is this property of continuous functions equivalent to anything familiar? If not, does it at least have a name?

If I understand correctly, every morphism of topological spaces $f : Y \leftarrow X$ factorizes uniquely into a composite of three morphisms $$f = c \circ b \circ a$$ such that $c : Y \leftarrow ...
-2
votes
0answers
46 views

Simplicial Sets

By definition a simplicial set is a functor from the simplex category $Δ^{op}$ to $Set$. But given an arbitrary set namely $S$, when we call this, a simplicial set? Can you give some examples please? ...
3
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0answers
38 views

The bigger picture the Five Lemma fits into

The Five Lemma is a statement in category theory about certain conditions under which certain maps in exact sequences are isomorphisms. It has a few relatives like the 4 lemmas and maybe the Nine ...
0
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1answer
38 views

Notation for kernel object

When $f: A \mapsto B$ is a morphism in some category with a zero object and limits, we can use $\ker(f)$ to refer to an equivalence class of morphisms to $A$ which satisfy a particular universal ...
2
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1answer
53 views

Projective sequence of C*Algebras by factors of embedded ideals isomorphic to algebra

Let $A$ be a $C^*$-algebra and $$A = I_1 \supset I_2 \supset I_3 \supset\ldots$$ be a sequence of embedded ideals in $A$ such that $\bigcap_{i=1}^\infty I_i = 0$. Is it true, that the projective limit ...
0
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1answer
57 views

Geometric Morphism

I am trying to understand the whole concept of toposes and how the geometric morphisms are involved to it. But always i stumble upon a new concept, something that i am not familiar with and a new ...
1
vote
1answer
36 views

Is an “$\aleph_0$-limit” a finite limit or a small limit?

I am sure this is a very trivial question. But I do not know anything about cardinals, and the nLab is full of them. I just want to know how to interpret a statement of the form "$\mathcal{C}$ has ...
1
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2answers
39 views

How to take a limit of a diagram with more than one category?

The German Wikipedia describes how one can define the quotient field of a ring over a universal property: A quotient field $(\mathrm{Quot}(R), i)$ of a ring $R$ is a field $\mathrm{Quot}(R)$ ...
5
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33 views

natural weak factorization systems

I am trying to understand the definition of natural weak factorization systems from this article by Tholen and Grandis, and these notes by Emily Riehl. In Riehl's notes, the splitting $s,t$ are of ...
0
votes
1answer
30 views

What can you do with a Frobenius Monad? [on hold]

A frobenius monad is an endofunctor that has natural transformations making it both a monad and a comonad. What are the most interesting aspects to Frobenius monads?
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1answer
86 views

Category of Sets with only monomorphisms

I came to work with a category which is the category of sets, "Sets", but for which I consider only arrows that are monomorphisms (i.e. injective maps). This makes sense in particular when expressing ...
1
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1answer
57 views

Reference/Definition of Homotopy in an Abstract Category

Let $\mathscr{C}$ be a complete and cocomplete category, and let $W$ be the collection of weak equivalences relative to some model on $\mathscr{C}$. We can form the homotopy category by localizing at ...
5
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0answers
28 views

Replacing faces of a cube in a quasicategory

I have a question on quasicategories which seems to be unavoidably cubical, and which I haven't been able to locate any information about. Suppose I have a cube $U:\mathbf{2}^3:\to C$ in a ...
4
votes
1answer
197 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
1
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1answer
33 views

Are arbitrary coproducts filtered colimits?

A coproduct in a category $\mathcal{C}$ is a colimit over a diagram $F:S\to\mathcal{C}$, where $S$ is a set. So my question is equivalent to asking whether a set is a filtered category. The nLab ...
0
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1answer
55 views

Representable Functors and Upper Sets (Final Segments)

I was looking around Wikipedia and came across this for representable functors: From another point of view, representable functors for a category $\mathcal{C}$ are the functors given with ...
0
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0answers
37 views

How to call a “non-strict” monoidal category?

A monoidal category is a category $\mathsf{C}$ equipped with a bifunctor $\otimes : \mathsf{C} \times \mathsf{C} \to \mathsf{C}$, a unit object, an associator, and right and left unitors satisfying a ...
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1answer
33 views

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) ...
0
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0answers
56 views

Is the linear operators must be invertible to from a category?

I am trying to understand the concept of category in mathematics. For example the following link talks about category $Lin$ which is an Abelian category. ...
0
votes
1answer
27 views

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 ...
1
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1answer
29 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 ...
0
votes
2answers
113 views

The Adjunction $\_\times A\dashv (\_ )^A$ for Preorders: The Deduction Theorem.

The following is from Turi's Category Theory Lecture Notes. Definition 11.11 Let $A$ be an object of a category $\mathbb{C}$ with binary products. The right adjoint of $\_\times ...
4
votes
2answers
93 views

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 ...
19
votes
3answers
2k views

What is the origin of the expression “Yoneda Lemma”?

Thank you very much in advance for telling where the expression “Yoneda Lemma” comes from. EDIT 1. On page -14 of Reprints in Theory and Applications of Categories, No. 3, 2003. Abelian Categories, ...
7
votes
1answer
198 views

Whatever Happened to Nearness Spaces?

I came across this paper about Nearness Spaces. It seemed to be at the time (1970-80s) a promising approach to general topology via category theory. I have found no posts at all on stackexchange ...
1
vote
1answer
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 ...
15
votes
3answers
405 views

Construction of a Hausdorff space from a topological space

Let $X$ be a topological space. Is there a Hausdorff space $HX$ and a continuous function $i:X\rightarrow HX$ such that for any Hausdorff space $A$ and a continuous function $j:X\rightarrow A$, there ...
1
vote
1answer
53 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 ...
3
votes
1answer
34 views

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 ...
4
votes
0answers
58 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 ...
5
votes
1answer
57 views

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. ...
1
vote
1answer
130 views

An equivalence of categories that is not adjoint. [duplicate]

Is there a good example of an equivalence of categories $F:\mathcal{A}\to\mathcal{B}$ such that $F$ is neither left nor right adjoint to its inverse $G:\mathcal{B}\to\mathcal{A}$?
19
votes
3answers
1k views

In category theory why is a right adjoint not a left adjoint?

I'm learning basic category theory and teaching myself about adjoints. The definition I have is that an adjunction between $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ is a ...
1
vote
1answer
54 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 ...
2
votes
0answers
37 views

Intuition for universal quotient maps [migrated]

The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman, the two characterizations of ...
2
votes
1answer
36 views

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 ...
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4answers
121 views

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 ...
0
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1answer
30 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 ...
4
votes
2answers
58 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!
4
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2answers
534 views

Small categories

Why $R$-Mod is a small category? There is a way to recognize small categories? For example Grp (i.e. category of all groups) is large because every set can be equiped with a group structure.
3
votes
0answers
32 views

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 ...
4
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0answers
41 views

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 $K$ ...