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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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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26 views

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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17 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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24 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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61 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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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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19 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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16 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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82 views

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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42 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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55 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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38 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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53 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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32 views

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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60 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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2answers
51 views

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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37 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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3answers
56 views

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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2answers
56 views

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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114 views

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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56 views

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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53 views

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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66 views

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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35 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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39 views

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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1answer
43 views

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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26 views

Subcategory of sets with surjective mapping [closed]

Im new at algebra and Im trying to prove that a subcategory of Set category , where the objects are sets and morphisms are surjective mappings is really a subcategory and it is not full. The first ...
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1answer
23 views

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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110 views

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
52 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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20 views

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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51 views

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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1answer
41 views

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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52 views

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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1answer
28 views

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
79 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 ...
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1answer
52 views

Constructing pullback and pushout problem

i) Let $p$ be a prime and $f: \Bbb Z \rightarrow \Bbb Z_p$ and $g: \Bbb Z_{p^2}\rightarrow\Bbb Z_p$ be the canonical epimorphism. Show that the pullback of $f$ and $g$ is isomorphic to $\Bbb Z ...
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50 views

Geometric Homotopy as Chain Homotopy

In Can we think of a chain homotopy as a homotopy, I learned that chain homotopy can be defined in an analogous fashion to homotopy, i.e from the product with an interval object etc. What about the ...
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238 views

Subcategories of $\mathbf{Top}$ which are closed under arbitrary products and coproducts

Is there a classification of those (full) subcategories of the category $\mathbf{Top}$ of topological spaces that are closed under arbitrary products and arbitrary coproducts in $\mathbf{Top}$? EDIT: ...
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Sequences or 'chains' of adjoint functors [duplicate]

Suppose we have (some categories and some functors such that) $F_1$ is left adjoint to $G_1$, $G_1$ left adjoint to $F_2$, $F_2$ left adjoint to $G_2$. Will $F_1$ then be equal to $F_2$ (and $G_1$ to ...
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About certain regular epimorphisms in a Grothendieck Topos

I am supposed to prove a rather technical property which should hold in any Grothendieck Topos, but I have troubles in accomplishing this task. Here is the context for the question. Let then ...
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49 views

Contravariant functor properties

What does F as an exact contravariant additive functor preserves or changes over an abelian category? (i.e kernels, cokernels, images, etc) Thanks
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37 views

Additive exact functors preserve homology of modules

If $C$ is a chain complex of modules over a ring $R$ and $F: Mod_R \to Mod_S$ an additive exact functor then: If $F$ is covariant then $H_n(FC)\cong FH_n(C)$. Can anyone give my an idea or an ...
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150 views

More than one pair of “nice” adjoint functors between different concrete categories

Though adjoint functors provide a universal description for many concrete mathematical constructions, these constructions usually revolve around finding a single "canonical" way to transform one type ...
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1answer
29 views

Proving distributivity of Heyting algebras with the Yoneda lemma.

How can one prove distributivity of a Heyting Algebra via the Yoneda lemma? I'm able to prove it using the Heyting algebra property $(x \wedge a) \leq b$ if and only if $x \leq (a \Rightarrow b)$. ...
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1answer
43 views

Category with pullbacks but not equalizers

Is there an example of a category with pullbacks but not equalizers (i.e. at least one pair of parallel morphisms does not have an equalizer)? Such a category cannot have have the terminal object, it ...
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What is the best path to learn Category theory and Type thoery?

I have little background in Programming in functional languages and wanted to learn type theory. I started with taking Homotopy type theory class Online videos of Robert Harper. I thought rather ...