# Tag Info

9

0) The excellent mathematician you evoke has as family name (=surname) tom Dieck and as first name Tammo: tom is part of his surname and has nothing to do with Tom, the endearing form of Thomas. 1) Your idea of "finding a book that starts with a formal treatment of the basics of category theory and moves to more advanced/specialized concepts in a ...

4

This process stabilizes imediately. To see this, note a somewhat more general result: if a diagram $D$ has an initial object $i$, then every functor $F : D → \mathscr C$ has a limit $Fi$ (with the obvious limiting cone). Given another cone $ΔC ⇒ F$, the component $C → Fi$ at $i$ is the required unique factorization through the limiting cone. In your case, ...

3

First of all, a diagram in $\mathcal{C}$ is not the same thing as a directed graph. A $\Gamma$-shaped diagram in $\mathcal{C}$ is a morphism of graphs $\Gamma\longrightarrow U(\mathcal{C})$, where $U(\mathcal{C})$ denotes the underlying directed graph of $\mathcal{C}$ ${}^{1)}$. Secondly, it makes sense to talk about commutativity for all kinds of graphs. ...

3

Not necessary, consider the directed graph on $A,B,C,B',C'$ with arcs $A\to B$, $B\to C$, $A\to C$, $A\to B'$, $B'\to C'$, and $A\to C'$. This directed graph is a counterexample to your claim since $A\to B$ and $A\to B'$ do not lead to a common vertex. P.S. Combinatoric is not a word, combinatorial is.

2

If $F,G:\mathcal A\rightarrow\mathcal B$ are functors and $\alpha:F\stackrel{\bullet}{\rightarrow}G$ is a natural transformation such that $\alpha_{A}:F\left(A\right)\rightarrow G\left(A\right)$ is invertible for each object $A\in\mathcal A$ then it can be shown that the inverses $\alpha_{A}^{-1}:G\left(A\right)\rightarrow F\left(A\right)$ are the components ...

1

In Functional Analysis, one often speaks of projective limits (limits in the category LCS of locally convex spaces) and inductive limits (colimits in LCS). In the latter case one has to be careful as colimits in TOP usually differ from those in LCS. What you state as a theorem says that every locally convex space is a projective limit of seminormed spaces. ...

1

As Zhen Lin said in the comments, there is a very general argument that answers your problem. Denote $\omega$ for the category of finite ordinals with set-functions between them. Definition 1. A Lawvere theory is a finite-product-preserving bijective-on-objects functor $\ell \colon \omega^\circ \to \mathcal T$. A morphism $f$ from the Lawvere ...

1

The category $(A\downarrow \mathcal{M}\downarrow B)$ is not necessarily a model category, as it can fail to be complete or cocomplete (or both). For example, $(\{*\} \downarrow \mathsf{Set} \downarrow \varnothing)$ is not a model category: it is empty! There is no map from a singleton to the empty set. However one can define, for every $f : A \to B$, the ...

1

Yes, as noted in comments, your argument works. If $K$ is a compact subset of $X$ and $U$ is open subset of $Y$, then the inclusion $A\subset Y$ maps $\{f\in C(X,A):f(K)\subset A\cap U\}$ to the $\{f\in C(X,A):f(K)\subset U\}\bigcap C(X,A)$. So the map $C(X,A)\to C(X,Y)$ identifies elements of subbase of own compact-open topology of $C(X,A)$ with elements ...

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