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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Is there a canonical functor F : 𝒞ᴼᴬ ⟶ 𝒞ᴬᴼ from the usual objects-and-arrows definition of a category to its arrows-only formulation?
Is there a canonical functor
$F : \mathcal{C}^{\mathrm{O\!A}} \longrightarrow \mathcal{C}^{\mathrm{A\!O}}$
from the usual objects-and-arrows definition of a category to its
arrows-only formulation?
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$\beta$ reduction rule for $\Pi$-types on free variables
The $\beta$-reduction rule for $\Pi$-types in dependent type thoery states that $(\lambda x:A.t)(u)=t[u/x]$ (provided $u:A$ in suitable contexts), which makes perfect sense to me.
However, due to some ...
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Mac Lane's characterization of bifunctors in terms of one-variable functors
I am trying to understand Mac Lane's characterization of bifunctors in terms of one-variable functors (proposition 1 in §1.3. of Categories for the Working Mathemathician). The theorem is as follows:
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On the definition of a Monoidal Category
In all sources I have read, a monoidal category is defined essentially as follows. A monoidal category is a 6-tuple $(\mathcal{C}, I, \otimes, \alpha, \lambda, \rho)$ where
$\mathcal{C}$ is a ...
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Duality and projectively stable category
Let $A$ be a finite‑dimensional algebra over an algebraically closed field $k$. I work with right modules.
We have a duality given by $F=\operatorname{Hom}(_,A):\mathrm{mod}(A)\to\mathrm{mod}(A^{op})$....
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Morphisms in $\mathfrak{R}^3$ and $\mathfrak{R}^6$ with $\mathcal{mod}\,A$
I am working with representation theory of finite-dimensional algebras.
While studying powers of the radical of the module category, I found the following
definition (for indecomposable modules):
$$
\...
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The role of strictness in the equivalence between $ 2 $-dimensional topological quantum field theories and Frobenius algebras
Monoidal categories and monoidal functors come in many flavors. The former can be weak or strict, while the latter can be lax, strong or even strict themself.
In it's Frobenius Algebras and 2D ...
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Additive Right Adjoint functor F preserves left exact sequence? Question about the proof
$f:A\to B$ and $g: B\to C$ are two morphisms of R-Mod. $0\to A \to B\to C\to 0$ is exact and F is an additive right adjoint functor.
I want to show $0\to F(A) \to F(B)\to F(C)\to 0$.
I know right ...
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Why are the projection functors in the definition of the product of categories explicitly defined?
Consider the usual definition of a product in a fixed category $\mathcal C$:
we have a family of objects (with projection maps) $\left\{\left( X_i, \, \pi_i \colon X \to X_i \right) \right\}_{i \in \...
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Supremum of Subsheaves
I'm reading MacLane and Moerdijk's "Sheaves in Geometry and Logic," and I'm having trouble understanding a description given in section 8 of chapter III of the supremum of a family of ...
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Doubt on how much is unique a certain homomorphism into the modules of differential
Let $k$ be a ring and let $A$ be a $k$-algebra. Denote by $\Omega^1_{A/k}$ the module of derivations on $A$ over $k$.
Let $I\subset A$ be an ideal. The composition of $k$-linear homomorphisms $$I\...
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split coequalisers which are not split epi [closed]
Any split epi is a split coequaliser. I assume the converse is not true, otherwise there will be no reason to have both concepts. However, I do not have a counterexample for this.
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$a \cong a$ using arrows-only axioms [closed]
This is an exercise in Topoi by Goldblatt:
$a \cong a$,
which means:
every object in a category $\mathscr{C}$ is isomorphic to itself.
A very nice way to think about this, in my experience, is to ...
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degeneracy maps when computing limits
I am reading page 15 of this note. It is stated that
May I ask, in the case of 2-limit, are the degeneracy maps needed to form the correct 2-limit?
(Perhaps a related question: In general, how do one ...
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Extending a mapping on objects to a functor in the category of Abelian Groups
The motivation for this problem is that it is a component in proving the Freyd-Mitchell embedding theorem.
Let $F$ be a functor from a small abelian category $C$ to the category of abelian groups $\...
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Characters of the multiplicative group
Fix a commutative, unital, associative ring $R$, and let $C$ be the category of commutative, unital, associative $R$-algebras.
Let $\mathbb{G}_m$ be the functor from $C$ to groups, which sends $A \in \...
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Is a retract a projective object in the category of TOP?
A retract of a top space $X$ is a subset $A\subset X$ s.t. there is a cts surjection $r: X\to A$ satisfying $r\circ \iota_A=Id_A$. The map $r$ is called a retraction and satisfies $r^2=r$.
Is every ...
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Class coincidence in sSet
Let $S$ be a class of simplicial sets such that
a) $S$ is closed under coproducts;
b) if in the pushout
$
\begin{array}{ccc}
A & \xrightarrow{f} & A' \\
\downarrow{g} & & \downarrow{h} ...
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Kernel zero but not monic?
Is there a category $\mathcal{C}$ with zero object and a morphism $f:A\to B$ such that $0:A\to A$ is a kernel but $f$ is not monic?
If $\mathcal{C}$ is preadditive then $\ker f = 0 \iff f$ is monic. ...
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meaning of A standard n-simplex may also be thought of as the “free simplicial set on a single n-cell”.
I'm studying Rezk's Introduction to Quasi-Categories and I don't understand the meaning of the sentence: “A standard
n-simplex may also be thought of as the free simplicial set on a single
n-cell.”
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Infinite internal direct product for groups
Given an arbitrary collection $\{A_i\}$ of groups, their direct product $A$ is defined to be the group together with the projections $A\longrightarrow A_i$ that satisfy the universal property for ...
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Proving arbitrary products exist in a Mac Lane universe
I am going through some leftover exercises from Mac Lane's Category Theory for the Working Mathematician. I am currently dealing with the foundational chapter, in which the concepts of small and large ...
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Conditions to be well-powered in an abelian category [closed]
My question refers to Lemma 1.14 on page 18 of Swan's Algebraic K-Theory. He seems to claim that if a category $B$ is abelian, complete, has enough projectives, and has a cogenerator, then $B$ is well-...
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Commutative cube for horizontal composition?
I'm using Emily Riehl's Category Theory in Context and these definitions and notes are in section 1.7, pages 45-46.
Definition 1.7.5 (Horizontal Composition): Given parallel functors $F, G: C \...
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How to show that $\operatorname{lim}_{n \in \mathbb{N}} R[x_0, \ldots, x_n] \cong \operatorname{lim}_{(A, p(t))} A$?
Let $R$ be a commutative ring with unity. Let $I$ be the following index category:
The objects of $I$ are pairs $(A, p(t))$ where $A$ is an $R$-algebra and $p(t) \in A[t]$ is a polynomial
A morphism ...
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When are the split morphisms given by the triangle identities isomorphisms?
Given an adjunction $(L,R, \eta, \epsilon)$ between categories $C$, $D$, the triangle identities tell us that the natural transformations $\epsilon L\colon LRL \to L$, $L \eta \colon L \to LRL$, $R\...
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Why a certain condition is satisfied in triangulated categories
Let $\mathcal T$ be a triangulated category and let $\cal S$ be a triangulated subcategory.
Say that a morphism $A\xrightarrow \alpha B$ is in $\Sigma\subset \mathrm{Mor}(\cal T)$ if, in the ...
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The skeletons of a subvariety of closure algebras form a subvariety of heyting algebras
Assume that $\mathrm{CA}$ be the category of closure algebras and $\mathrm{HA}$ the category of heyting algebras both as varieties (or equational classes). Let $\Delta: \mathrm{CA} \to \mathrm{HA}$ be ...
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How to show that if $D \hookrightarrow X$ is bold and $F$ is a sheaf, then $\operatorname{Hom}(X, F) \to \operatorname{Hom}(D, F)$ is a bijection?
Let $\mathcal{C}$ be a small category.
$\operatorname{PSh}(\mathcal{C}) = \operatorname{Fun}(\mathcal{C}^{\operatorname{op}}, \operatorname{Set})$ is the category of presheaves on $\mathcal{C}$.
For ...
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Is the underlying vector space of a condensed $\mathbb{R}$-vector space a topological $\mathbb{R}$-vector space?
According to the answer given at What exactly is a condensed $\mathbb{R}$-vector space? , a condensed $\mathbb{R}$-vector space is an internal module object in $\operatorname{CondSet}$ over the ...
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What exactly is a condensed $\mathbb{R}$-vector space?
In Peter Scholze's lecture notes Lectures on Analytic Geometry , the fundamental objects of study are the so-called condensed $\mathbb{R}$-vector spaces. However, as far as I can see, no definition of ...
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Can “points” of objects be defined without assuming the category has a terminal object?
For a category $\mathcal{C}$, for an object $C$ in $\mathcal{C}$ call a morphism $p: C \to C$ a “(generalized) point” of $C$ if
$p$ is idempotent, i.e. $p \circ p = p$.
$p$ is “maximally non-mono”, i....
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Is a functor a "natural function" between morphisms?
Let's consider only small categories for simplicity.
Let $C$ and $D$ be categories, $F_0 : C_0 \to D_0$ be a function and $F_1 : C(x,y) \to D(F_0(x),F_0(y))$ be a family of functions indexed by ...
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Understanding lax natural transformations and adjunctions between strict 2-functors
I'm relatively new to higher category theory, and although I am familiar with "regular" category theory, I am having some trouble understanding the terminology and definitions used here.
For ...
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How to glue the natural transformations?
I encountered a question when I was reading an article about the $p$-adic half plane. But I think this question do not require the knowledge about it.
Let $O$ be a DVR with a uniformizer $\pi$. We ...
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Error or imprecise statement in Mac Lane's "Categories for the Working Mathematician"?
$\DeclareMathOperator{\Nat}{Nat}\DeclareMathOperator{\Map}{Map}\DeclareMathOperator{\Ob}{Ob}\DeclareMathOperator{\Set}{Set}\DeclareMathOperator{\id}{id}$Let the definitions and assumptions in set-...
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Two Functors $\mathsf{Grp} \to \mathsf{Grp}$ Leaving the Objects.
I'm new to category theory, and this question popped up from an exercise in MacLane's "Categories for the Working Mathematician." The exercise is:
Exercise I.3.5
Find two different functors ...
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Cartesian product interpretation of transfinite numbers up to $\epsilon_0$ as sets of tuples; difficult to compose $-$, $/$
Interpreting expressions of $k\in\mathbb{N}, \omega, $ and $\text{+ - */^}$ as types of tuples
Section 1 - Preamble
The ordinal-like expressions tend to cause confusion, so in this question I define ...
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Matrix category and category of finite-dimensional vector spaces equivalent as categories, but proof invokes choice
This exercise comes from section §1.4 in Mac Lane. I will briefly state the problem, following by my solution and a question.
First consider a commutative ring $K$. The category $\mathbf{FinVec}$ ...
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left and right units are homotopic for idempotent associative algebras
First let me recall two facts:
$\bullet$ Consider an idempotent associative algebra $A$ in some monoidal category $\mathcal D$. We have a unit $\eta:1_{\mathcal D}\to A$. Suppose a multiplication map $...
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Doubt on left Ore conditions
Let $\cal C$ be a category and let $\Sigma$ be a class of morphisms in $\mathrm{Mor}\cal (C)$. Those are the left Ore conditions given in my course:
given morphisms $f:x\to y$ and $g:y\to z$, if at ...
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being a module over an idempotent algebra is a property
I hear the slogan "being a module over an idempotent algebra is a property". Formally I guess that it means that if
$\bullet$ $A$ is an idempotent $R$-algebra i.e. the multiplication map $A\...
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In the free topos $\mathbb{A}^1$ on an object $X$, what is $\operatorname{Hom}_{\mathbb{A}^1}(X, X)$?
Let $\mathbb{A}^1 = \operatorname{PSh}(\operatorname{FinSet}^{\operatorname{op}})$ be the free topos on a point. This has the universal property that for any topos $\mathscr{X}$, we have
$$
\...
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Trying to understand the arrows-only description of a natural transformation, but cannot form the composition $\tau(g) \circ \tau(f)$
It is an elementary result that a natural transformation $\tau \colon S \to T$ defines a function (also called $\tau$) such that for each composable pair $(g, f)$ of morphisms we have $T(g) \circ \...
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Removing the asymmetry between colimits and limits in presentable categories
As discussed here Asymmetry between colimits and limits in presentable categories, there is an asymmetry in the definition of presentable categories that prefers colimits over limits. The definition ...
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Question about example of free object as universal object
I came across the following example in Algebra by Hungerford (1974, 57-58):
“Let $F$ be a free object on the set $X$ (with $i : X \to F$) in a concrete category $\mathcal{C}$. Define a new category $\...
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Asymmetry between colimits and limits in presentable categories
I know that a colimit in $C$ is equivalently a limit in $C^{op}$, which gives an equivalence $op: Cat^{(small) colimits} \cong Cat^{(small) limits}$ between categories with colimits, limits and ...
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Is there an adjunction of the form $[\mathcal{C}^{\text{op}}, \text{Set}^{\text{op}}] \leftrightarrows [\mathcal{C}^{\text{op}}, \text{Set}]$?
Let $\mathcal{C}$ be a category.
Consider the following four functors:
$$
\newcommand\op[1]{{#1}^{\operatorname{op}}}
\newcommand\Set{\operatorname{Set}}
\newcommand\Hom{\operatorname{Hom}}
\...
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Doubt on a non canonical isomorphism in exact functors
Let $\mathcal T$ and $\mathcal S$ be triangulated categories (I'm using the definition of triangulated category given here, because that is the definition that my teacher uses, even if I have seen ...
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Is the geometric realisation of a simplicial category a topological category?
Let $X$ be a simplicial set. Then we may consider its geometric realisation $|X|$, which is a topological space. What would happen if $X$ were not only a simplicial set, but also a simplicial (small) ...
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