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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What is a “category”?

I have tried to understand category theory for a while but never been able to get it. I finally found a text I like called Category Theory for Scientists. However, I think it would be easier to read ...
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Functorial Properties Preserved by Natural Transformation

This question was born from (and is in a sense a continuation of) this one, about functorial properties preserved by natural isomorphisms. What functorial properties are preserved by natural ...
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Functorial Properties Preserved by Natural Isomorphism

Conceptually, functors which are naturally isomorphic should have the same functorial properties e.g exactness, (co)continuity, etc. Thus, ideally, I'd hope for a precise definition of a functorial ...
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24 views

Tensor-Hom Adjunction In Monoidal Categories?

Is there a generalization of the tensor-hom adjunction to monoidal categories, or is it a special property of $\mathsf{Mod}$-$R$?
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36 views

Adjunctions via Reflections and the Axiom of Choice

I have met two ways of defining adjunctions: via the triangle identities, and via reflections. Proposition 3.1.2 Let $F:\mathsf A \rightarrow \mathsf B$ be a functor and $B$ an object of $\mathsf ...
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Non-Universal Delta Functors

Recall a delta functor on an abelian category is a collection of functors $H^n$, $n \ge 0$ and connecting homomorphisms associated to each short exact sequence so that we get a corresponding long ...
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28 views

How to calculate number of elements in HomSet

Im giving category theory a chance but have very limited math background, I'm learning from the book "Category theory for the sciences" but got lost on page 16 :) Exercise 2.1.2.12. Let ...
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40 views

Preferred way to write elements of the direct sum of vector spaces

Suppose $V$ and $W$ are vector spaces over the same field and $V\oplus W$ is their direct sum. Reading through the literature I found essentially two ways of writing elements of $V\oplus W$. 1.) We ...
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Motivation for Definition of Derived Category

On the $n$Lab entry about derived categories, I read the derived category of an abelian category $\mathsf A$ is the localization of $\mathsf{Ch}_\bullet (\mathsf A)$ at the quasi-isomorphisms. My ...
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39 views

Quick question: G-set functor

The Wikipedia page on Representable Functor says: A group G can be considered a category (even a groupoid) with one object which we denote by •. A functor from G to Set then corresponds to a ...
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97 views

Questions about a topological category

Given topological spaces $(X_i,\tau_i)$ with sets $\mathbf S_i=\{\mathcal A\in \tau_i^2|\mathcal A\supseteq\Delta X_i^2\}$, where $\tau^2$ is the product topology and $\Delta$ is the diagonal. The ...
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inverse limit in the plane

What stuff can I say about inverse limits regarding the mapping of $[0, 1]$ onto $[0,1]$ given by $$f(x) = \left\{ \begin{array}{ll} 2x & \mbox{if } 0 \le x \le {1\over2}\\ 1 & \mbox{if ...
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40 views

Group objects in the category of rings

Are there group objects in: $\text{Ring}$ $\text{CRing}$ If so, why doesn't anyone talk about them? On the other hand, $$ \begin{align} cogroup \ objects \ in \ \text{CRing} &= co(group \ ...
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Functors that has a natural transformation from identity

Let $F:\mathcal{A}\to\mathcal{A}$ be a functor with a choice of $A\to F(A)$ for every $A\in \operatorname{Ob}\mathcal{A}$, such that $$\require{AMScd} \begin{CD} A @>{f}>> B\\ @VVV @VVV \\ ...
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1answer
34 views

Simple question on diagrams in a category

Forgive the simplicity of my question but after running across the definition of a diagram in $C$ of shape $J$ as simply a functor $D:J\rightarrow{C}$ does this require $J$ to be a subcategory of $C$ ...
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34 views

Product / Co-product in category of sets and relations (Rel) [on hold]

What is the co-product and product in category of sets and relations. Thanks
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68 views

Continuity of Modified Hom Functor

I have been studying category theory and have been exploring hom functors. I've come across an interesting question and after spending several hours thinking about it, haven't gotten anywhere. Let ...
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2answers
67 views

A Tool to practice Categories / Allegories

Is there any handy tool to practice Categories / Allegories, in the sense that for a defined Category, it is possible to check the result of an operation application. For example, a tool which ...
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0answers
23 views

A ``partial'' Mitchell-Benabou language?

I am investigating the category in which there happens to be no subobject classifier for the particular way in which it is formulated. But, there is an object however who is very close to a subobject ...
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18 views

Subobjects Equivalent iff Isomorphic Domains?

Regarding subobjects as monics (not as equivalence classes of monics), I seem to have proven that subobjects are equivalent iff their domains are isomorphic as objects of the category in question. ...
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1answer
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Is my understanding for group algebra correct?

Let $G$ be a group, $k$ be a commutative unital ring. Consider $\mathbf{Alg}_k^{inv}$ the category of unital $k$-algebras whose multiplicative semigroup is a group. Then there is a forgetful functor ...
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68 views

The tensor product of monads.

It is known that the tensor product of endofunctors End(C) over a given category C is given by composition and the category of monads Mon(C) over a given category is cartesian. That cartesian product ...
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1answer
66 views

Exactness and Naturality

I'm trying to read this blog post about exact functors, and I see mentions of naturality which I have not stumbled upon elsewhere. In particular, in the proof of the Theorem, the author says By ...
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41 views

Quotient Objects in $\mathsf{Grp}$ II

This question is a sort of continuation of a previous one. In CWM, Maclane says ... every quotient object of a group $G$ in $\mathsf{Grp}$ is represented by the projection $\pi:G\rightarrow G/N$ ...
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59 views

Categorical Interpretation of Localization

At the very beginning of Ravi Vakil's amazingly famously amazing and famous notes on algebraic geometry, he remarks that some familiarity with localization and prime ideals is useful. I don't know ...
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A More Advanced Version Of Aluffi

Paulo Aluffi's Book, Algebra, Chapter 0 aims to teach basic algebra from a categorical viewpoint. The first chapters of the book, however, introduce groups and rings using very basic categorical ...
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Kernel in additive category

Supose $\mathcal{C}$ an additive category. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ and additive functor. If $k: K \rightarrow X$ is the kernel of the morphism $f: X \rightarrow Y$ and we have ...
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62 views

Given a category with exponentials, $\lambda f \circ g = \lambda (f \circ (g \times id))$

I'm trying to prove the next lemma, and I can't seem to find a solution, even though it looks pretty easy. The lemma is, given a category with exponentials, $\lambda f \circ g = \lambda (f \circ (g ...
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Trying to define a functor with exponentials

I need to define a functor $F:C \rightarrow C$ that maps $X$ to $R^X$, where $R$ is a fixed object. The thing is, given an arrow $f:X \rightarrow Y$, $F(f)$ has to be an arrow from $R^X$ to $R^Y$, but ...
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1answer
38 views

A Lemma from Freyd

This is a lemma from Freyd's Abelian Categories stated without proof. In an abelian category, $$A\rightarrow S \rightarrowtail B = A \rightarrow B$$ if and only if $$A\rightarrow B \twoheadrightarrow ...
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Reference for $(\infty,1)$-Categories

I am looking for an organized source from which I can learn about $(\infty,1)$-categories. I am unable to learn the concept from the $n$lab alone. Here it is said that Lurie called ...
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3answers
55 views

Quotient Objects in $\mathsf{Grp}$

I don't know how to precisely formulate my question, but here goes: Subobjects and quotient objects are duals, so a quotient object in $\mathsf{Grp}^{\text{op}}$ is a subobject in $\mathsf{Grp}$. The ...
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Terminology question: what does a natural isomorphism do to maps?

Suppose I have categories $C$ and $D$ and naturally isomorphic functors $F,G\colon C \to D$. (I do. Trust me.) Now name the natural isomorphism $\theta$; then for any arrow $f\colon x \to y$ in $C$, ...
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Freyd: “is a subobject of” is not transitive

On page 20 of Abelian Categories, Freyd writes Note that the relation "is a subobject of" is not transitive. On page 91 of Awodey's Category Theory (there are several typos in this page; the ...
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Examples of applications of category theory to chemistry

What is some simple application of category theory to chemistry, namely, something that is much easier to do in chemistry with category theory than without. It does not need to be bleeding edge, or to ...
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Basic Axiomatic Definitions in Categories and Allegories (Freyd and Scedrov)

I'd like to ask about the Basic Definitions given at the very beginning of the Categories and Allegories. Some aspects of the text are idiosyncratic, so first I'll quote from the text: 1.1 BASIC ...
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1answer
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Sets: why slice category is not isomorphic to functor category

Is known that the slice category $\mathbb{Set}/I$ is equivalent to the category of $I$-indexed sets $\mathbb{Set}^{I}$. We can establish two functors $$\varphi: \mathbb{Set}^{I} \rightarrow ...
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1answer
38 views

isomorphism of pointed sets

What is an isomorphism in the category of pointed sets? Is it just an exact sequence $$ 1 \to A \to B \to 1 ?$$ (Note: even though the kernel of the middle map is zero, $A$ might not inject into $B$.) ...
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How can I define this natural transformation $X^{A+B} \to X^A \times X^B$?

So, I have to define a natural transformation between functors $F,G:C\to C$, where $F(X)= X^{A+B}$ and $G(X)=X^A\times X^B$. I figured out that $\alpha_X:X^{A+B}\to X^A\times X^B$ must be $\langle f,g ...
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Inverse image of a functor

Suppose $F: \mathcal A \to \mathcal B$ is a functor. We can define a category $F^{-1}(\mathcal B)$ as follows: an object is an object of $\mathcal A$, and a morphism between objects $A_0$ and $A_1$ ...
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Naturality of Transformations

When we say some arrow $\eta _A$ is natural in $A$ ($A$ being an object of the category in question, $\mathsf C$), we mean it is a component of a natural transformation. I have consistently stumbled ...
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Equivalence of categorical coproduct proof

quiLet $C$ be an abelian category and {$X_1$,...,$X_n$} a finite family of objects in that category. ( $X$,($M_i$: $X_i$$\to$ $X$) where $i_1$=1,....n a coproduct of the finite family if and only if ...
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Colimit in the category of (all) simply transitive group actions

Let $\mathcal{C}$ be the category of all group actions, i.e. : the objects are the pairs $(G,F)$ where $G$ is a group and $F$ is a functor $F\colon G\to\mathbf{Sets}$ a morphism between $(G_1,F_1)$ ...
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1answer
53 views

Definition of a group object in category theory

I have a question about the definition of a group object in Category Theory (see the Wikipedia article). $G$ is an object of $\mathcal{C}$, and we assume that $\mathcal{C}$ admits finite products. ...
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Simple problem about morphism in abelian categories

$f$ : $X\to$ $Y$ and $g$ : $Y\to$$Z$ a sequence in abelian categories. Show that if $gf$=$0$ if and only if exist a monomorphism $h$:$Im(f)$ $\to$ $Ker(g)$ such $kh$=$j$, where $j$:$Im(f)$$\to$ $Y$ ...
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Why is the coproduct in $\mathsf{Grp}$ so different from the coproduct in $\mathsf{Ab}$?

Why is the coproduct in $\mathsf{Grp}$ so different from the coproduct in $\mathsf{Ab}$? What about $\mathsf{Grp}$ makes for a seemingly far-more-complicated coproduct? If your answer revolves around ...
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Module of differentials in the functorial approach to schemes and quasi-coherent modules

Recall that for a functor $X : \mathsf{CAlg}(R) \to \mathsf{Set}$ from commutative $R$-algebras to sets one can define quasi-coherent $\mathcal{O}_X$-modules as "compatible" families of $A$-modules ...
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If $f= \mathrm{ker}\,g$, then $g = \mathrm{coker}\,f$?

I didn't understand a step in the proof of Proposition 5.92 from Rotman's Introduction to Homological Algebra (2nd Ed.) where he says: "there is a morphism $g: B\to C$ [in a given abelian category ...
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Direct limits in the category of modules

STATEMENT: Proof. Let $(I,≼)$ be a directed set, and let $\left\{Mi\right\}i∈I$ be a directed system of R-modules, with $\left\{f_{ji}\right\}_{i∈I},i≼j$ a corresponding directed family of ...
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1answer
86 views

“Internal” and “external” in maths, and also in vector spaces

I have looked at 3 books and it is clear that "internal" and "external" are two styles of defining something, I would like to know what they mean "generally" - that is very soft but it is clear to me ...