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Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure preserving morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seemingly different areas of mathematics.

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Mac Lane Chapter 7 Section 2 Exercise 1

Let $\mathcal{C}$ be a monodical category, with the monodical product written $\otimes$, the associator denoted $\alpha$, and the left/right unitors denoted $\iota^\ell,\iota^r$ respectively. Mac ...
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Are all unions in a topos with complete subobject lattices secretly colimits? On a logical analogue of the AB5 axiom

To clarify, here “topos” always means an elementary topos; I do not assume sheaves on a site, where I already knew my question to have a positive answer. It is known but not so immediate from the ...
FShrike's user avatar
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Majority vote EM category

I am working with the multiset monad. I want the EM category to have a structure map that is majority vote. That is to say that the map $f: X \rightarrow M(X)$ takes the set element with the highest ...
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Need Help Proving that $\mathbb{R}$ is a cogenerator of $\textbf{Vect}$

I am trying to prove that $\mathbb{R}$ is a co-generator of $\textbf{Vect}$. Recall that $B'$ is a co-generator of a category $\mathscr{B}$ is the collection of arrow $f:B \rightarrow B'$ is mono, for ...
babu's user avatar
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Need Help To Prove Fullness of a Functor

I am working on proving the following proposition A functor $F:\mathscr{A} \rightarrow \mathscr{B}$ is an equivalence of categories if and only if it is faithfull, full and essentially surjective. ...
babu's user avatar
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Alternative form of Kolmogorov extension theorem on filtered space

In an thesis HERE the author proved (categorically) an alternative form of Kolmogorov extension theorem as follows (p. 129~p. 130): let $I$ be an nonempty set, $(\Omega,(\mathcal{F}_i)_{i\in I}, \...
Westlifer's user avatar
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Suppose that $0\to A\xrightarrow{f} B \xrightarrow{g} C$ is exact in an abelian category. Prove that $0\to A\xrightarrow{f} B\to Im(f)\to 0$ is exact

Suppose that $0 \to A \xrightarrow{f} B \xrightarrow{g} C$ is exact in an abelian category. I am trying to prove that $0 \to A\xrightarrow{f} B \to Im(f)\to 0$ is exact. Here, I use the definitiom ...
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Infinite coproduct in a category of groups [duplicate]

Let $\mathbf{Grp}$ be a category of groups. Then, we know that there exists a coproduct $\bigsqcup_{I\in I} G_i$ for a family of groups $\{G_i\}_{I\in I}$ in $\mathbf{Grp}$ when $I$ is a finite set. ...
Yos's user avatar
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The notion of fundamental cogroup

I know there is a functor $\pi_1$ from the category of pointed topological spaces to the category of groups, sending each pointed topological space to its first fundamental group. I know that a group ...
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Short exact sequence of direct sums

Suppose that $R$ is a valuation ring with field of fractions $F$. Call an $R$-module nice if it is isomorphic to $X/Y$ for some $Y<X\leq F$, and suppose that whenever $$0\to K\to M\to N\to 0$$ is a ...
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Name of an endo-"functor" but which doesn't change the source/target of morphisms?

An endofunctor maps objects $A$ to $F(A)$ and morphisms $m:A\to B$ to morphisms $F(m):F(A)\to F(B)$. Is there an established name for a different "endofunctor-like" class of objects (not ...
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Continuous functors and fibrations

For ordinary fibrations is it true that: Given a continuous functor $F \colon C \to D$ with $C$ finitely complete, and a fully faithful functor $U$ such that $F \dashv U$,then $F$ is a fibration?
Makhulukhulu's user avatar
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Prove that any preadditive category can be embedded into an additive category as a full subcategory

I am trying to prove that any preadditive category $\mathcal{C}$ can be embedded as a full subcategory into an additive category $\mathcal{D}$. Since an additive category is, by definition, a ...
Squirrel-Power's user avatar
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Same morphism between more couples of objects?

I was studying elementary category theory when a question came to my mind. Is it possible in a category for the same "entity" to be a morphism between more than one couple of objects (i.e. ...
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Two functors having the same right adjoint functor. [duplicate]

Let $F,G:C\to D$ be two functors. Suppose that $F$ and $G$ have the same right adjoint functor, does this imply that $F=G$? I know that $F^*=G^*$ implies that for all $c\in C$ and all $d\in D$ we ...
Catalio13's user avatar
2 votes
1 answer
58 views

Understanding inverse image functor in terms of the adjoint universal property

Vakil (in his Rising Sea Algebraic Geometry text) "defines" the inverse image functor as the adjoint of the pushforward functor. Then he asks you to solve certain problems below using this ...
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Proof that quasi-isomorphisms form a localizing class in the homotopy category of complexes

I'm currently reading Gelfand and Manin's book Methods of Homological Algebra. Theorem III.4.4 says that the class of quasi-isomorphism in a homotopy category of complexes is localizing and I am ...
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3 answers
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What is the product in the category of sets with only the injections as maps?

What is the product in the category of sets with only the injections as maps? In the category of sets with all the functions/maps, the product will be the "$\times$", but in the subcategory ...
Lukarios 5157's user avatar
-1 votes
1 answer
50 views

Understanding slice category [closed]

I am self learning about category theory and I have a question about slice categories. Lets say we have a finite category C like so: a -> b -> c <- d. Let’s consider the slice category C/c. ...
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1 answer
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In the category defined by $\le$ on $\mathbb{Z}$, every morphism defined is a monomorphism and an epimorphism

I am studying with Aluffi's "Algebra: Chapter 0", and got stuck on proving the statement on pg. 30 here: ... in Set, a function is an isomorphism if and only if it is both injective and ...
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1 answer
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Morphism between product types as a product of diagonal maps

I am trying to follow this post https://ncatlab.org/toddtrimble/show/Notes+on+predicate+logic and I got confused on this paragraph: From this point of view, a morphism between product types is a ...
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Are comma categories over categories with pullbacks semi-secretly preorders?

Here is exercise IV.5.3 of Mac Lane's CWM: For $C$ a category with pullbacks, each arrow $f: a \to a'$ defines a functor $(C \downarrow a) = f_* : (C \downarrow a) \to (C \downarrow a')$ which carries ...
Mark's user avatar
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1 answer
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A sort of Day convolution without enrichment

Some time ago I was trying to define a monoidal structure on a functor category $[\mathcal{C},\mathcal{D}]$ between two monoidal categories $\mathcal{C}$ and $\mathcal{D}$, such that the monoid ...
Captain Lama's user avatar
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What morphism is sent to a monomorphism by the left Kan extension ${\rm Lan}_{\Delta}\colon{\bf Set^\Delta\to\bf Set^{\hat\Delta}}$ along Yoneda?

For any small category $C$, let us write $\hat{C} = \mathbf{Set}_C$ for the presheaf category $\mathbf{Set}^{C^{\mathrm{op}}}$, and $y=y_C\colon C\to \mathbf{Set}_C$ for the Yoneda embedding. Consider ...
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Proof that the Yoneda embedding is cartesian closed

Is there a reference somewhere that the Yoneda embedding is cartesian closed? I tried showing this myself, but after staring at it for an afternoon, I did not yet see a proof. I saw an answer here ...
Tempestas Ludi's user avatar
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Bayesian vs frequentist difference beyond interpretation from the algebraic or categorical standpoint

Context: I'm not a statistician at all, and I was just involved in a debate between physicists on fundamental differences between the bayesian and frequentist approaches. Someone was arguing that it's ...
Vincent's user avatar
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1 answer
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The core of an $\infty$-category and pointwise invertible maps $\Delta^1 \times \partial \Delta^n \cup \{1\} \times \Delta^n \to \underline \hom(B,X)$

I'm currently working through Theorem 3.5.11 of Cisinski's Higher Categories and Homotopical Algebra. The section in which the theorem is inscribed concers the core of an $\infty$-category. Trying to ...
qualcuno's user avatar
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1 vote
1 answer
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Inverse and Composition of Bisimulations

Exercise 63 of Rutten's The Method of Coalgebra: exercises in coinduction asks us to prove that "the collection of all bisimulation relations between two given stream systems is closed under (i) ...
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1 answer
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A full embedding functor from $Sob$ to $Frm$?

In the following passage, the authors Picado and Pultr say the functor is a full embedding then they say that it would be a full embedding under some condition. Any explanation?
Catalio13's user avatar
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Does $\varinjlim_{I_0\subset I\subset I_1} \operatorname \bigoplus_{i\in I} M_i=\prod'_{i\in I_1} M_i$ hold? Def of $\bigoplus$ and $\prod'$ [closed]

Let $M_i$ be set of abelian groups.Let $I_0$ be a fixed finite directed set.Let $I$ be a fixed infinite direct set which contains $I_o$. Does $\varinjlim_{I_0\subset I\subset I_1} \operatorname \...
Pont's user avatar
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2 votes
0 answers
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Studying modal logic using category theory

When reading about modal logic, namely general frames and algebras, I've been seeing a lot of potential functors and/or universal properties. Like constructing an algebra with Boolean operators from a ...
tses's user avatar
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10 votes
2 answers
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Nontrivial advantages of thinking of groups as groupoids with one object?

There are already a number of questions here asking "what do people mean when they say a group is a groupoid with one object?". A natural question to ask is "are there any interesting ...
Shrugs's user avatar
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1 answer
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What is the difference between direct sum $\bigoplus_{i\in I} M_i$ and restricted product $\prod'_{i \in I}M_i$ of abelian group?

What is the difference between direct sum and restricted product of abelian group? I recently heard that direct limit of direct product is not a direct sum but a restricted product. Until now, I ...
Pont's user avatar
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6 votes
1 answer
78 views

Category of Sets vs Category of Types (& Yoneda Lemma)

I've been wondering for a while why we discuss the category of sets, $\mathbf{Set}$, all the time, but hardly discuss something like the category of all types and functions between them ($\mathbf{Type}...
SFDK's user avatar
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4 votes
1 answer
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Higher Coherences and Maps from Colimits

In higher category theory there is the important mantra that commutativities are additional data and not just a property as in $1$-category theory. So the following question I'm proposing will ...
Qi Zhu's user avatar
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3 votes
0 answers
27 views

Is every balanced regular category exact?

A finitely complete category is said to be regular if coequalizers of kernel pairs exist and regular epimorphisms are stable under pullback. Also, a regular category is said to be exact if every ...
Geoffrey Trang's user avatar
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0 answers
54 views

Is there "meta" algebra, that operates with the collections of expressions?

Classical decision problem (https://web.eecs.umich.edu/~gurevich/Books/00.pdf - book, it opens as pdf) considers the sets of formulas which have decidable SAT and which have not decidable SAT problem. ...
TomR's user avatar
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5 votes
1 answer
121 views

Natural transformation picking out the map from the initial object

As so often, I'm failing to construct a map in $\infty$-category theory that is easily constructed in $1$-category theory. Let $\mathscr{C}$ be an $\infty$-category and let $F: \mathscr{C} \to \mathsf{...
Qi Zhu's user avatar
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2 votes
3 answers
51 views

Forgetful functor $V: \underline{\mathbf{PSet}} \rightarrow \underline{\mathbf{Set}}$ is not full

This might be a trivial question, but I don't understand why the trivial functor $V: \underline{\mathbf{PSet}} \rightarrow \underline{\mathbf{Set}}$ is not full. ($\underline{\mathbf{PSet}}$ is the ...
Minerva's user avatar
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1 answer
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On the definition of monomorphisms in arbitrary categories

I am used to working with the following definition of monomorphisms. Definition 1. In a category $\mathcal{C}$ we say that a morphism $f\in\operatorname{Hom}_{\mathcal{C}}(X,Y)$ is a monomorphism if ...
James Turesson's user avatar
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0 answers
44 views

Number of isomorphisms between categories

Let $\mathcal{C}_1$ be a category with objects $A=\{1,2,3\}$ and $B=\{4,5,6\}$. The number of isomorphisms between them is $3!=6$ and morphisms $27$. Can we extend this notion of number of ...
DatBoi's user avatar
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0 votes
1 answer
90 views

Can individual topological space be considered as category?

Of course, I am aware of Top (category of topological spaces). My question is about something different - can any topological space be considered as category? E.g. its objects may be the open sets of ...
TomR's user avatar
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3 votes
1 answer
76 views

Stable categories are tensored over spectra

It's a folklore result by Lurie that for a (presentable) stable $\infty$-category $\mathscr{C}$ the mapping space functor $\mathsf{Map}:\mathscr{C}^{\mathrm{op}} \times \mathscr{C} \to \mathsf{An}$ ...
Qi Zhu's user avatar
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3 votes
1 answer
71 views

Commutativity of second square when first square and outer rectangle commute

The general question I reach is with the diagram below $\require{AMScd}$ \begin{CD} A @>f>> B @>\pi>> C \\ @| @VbVV @VcVV \\ A' @>f'>> B' @>\pi'>> C' \\ \end{CD} ...
George's user avatar
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2 votes
1 answer
45 views

Compatibility of adjunctions for closed monoidal category

If $\mathcal V$ is braided monoidal closed, meaning that for any object $A$ the functor $-\otimes A$ admits a right adjoint $[A,-]$, then $\mathcal V$ is enriched over itself by letting $\mathcal V(A,...
Nikio's user avatar
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0 answers
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Concrete example on the commutative diagram of comparison functor

I am trying to familiarize myself with the properties of monad. I discovered in nlab the following convoluted picture about the whole story. https://ncatlab.org/nlab/show/comparison+functor It is ...
Y.X.'s user avatar
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1 vote
1 answer
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Models of a slice of coherent category

I am trying to get some sense of the proof of Theorem 10 in this link. https://www.math.ias.edu/%7Elurie/278xnotes/Lecture6-Completeness.pdf Immediately after it, Exercise 11 says a model of coherent ...
Y.X.'s user avatar
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1 vote
0 answers
35 views

The stalk of a presheaf is isomorphic to the stalk of its inverse image

Let $f:X\to Y$ be a continuous map of topological spaces, and let $\mathcal O$ be a presheaf of (commutative unitary) rings over $Y$. I'm trying to understand how the fact that colimits commute with ...
Ezio Greggio's user avatar
1 vote
1 answer
206 views
+50

A diagram with triangles and squares commute $\iff$ all triangle and square subdiagram commute

Intuitively, I want to show that if a commutative diagram is composed of only squares and triangles, then it commutes if and only if all squares and triangles sub-diagram commute, this is asked in ...
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4 votes
0 answers
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Do we know of a pair of categories for which we do not know if they are equivalent?

There are many categories we obviously know to be inequivalent due to some category-theoretical invariant one has that the other doesn't (for example, $\textbf{Grp}$ has a $0$ object while $\textbf{...
kabel abel's user avatar

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