Categorical jokes, tidbits, and factoids

4 minute read

Published:

\[\newcommand{\rv}[1]{\mathbf{#1}} \newcommand{\x}{\rv x} \newcommand{\y}{\rv y} \newcommand{\bar}[1]{\overline{#1}} \newcommand{\wtil}[1]{\widetilde{#1}} \newcommand{\what}[1]{\widehat{#1}} \newcommand{\ep}{\varepsilon} \newcommand{\ph}{\varphi} \newcommand{\maps}{\colon} \newcommand{\to}{\rightarrow} \newcommand{\xraw}{\xrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\To}{\Rightarrow} \renewcommand{\dot}{\centerdot} \renewcommand{\tensor}{\otimes} \newcommand{\pr}{^\prime} \newcommand{\op}[1]{#1^{\mathrm{op}}} \newcommand{\hom}{\textit{Hom}} \newcommand{\oo}{\circ} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\str}[1]{#1^{*}} \newcommand{\intvl}{\mathbb{I}} \newcommand{\yon}{\mathcal{Y}} \newcommand{\topos}{\mathfrak{Top}} \newcommand{\bTopos}{\mathfrak{BTop}} \newcommand{\BTop}{\mathfrak{BTop}} \newcommand{\con}{\mathfrak{Con}} \newcommand{\pos}{\mathfrak{Poset}} \newcommand{\topl}{\mathbb{Top}} \newcommand{\grpd}{\mathbb{Grpd}} \newcommand{\ab}{\mathbf{Ab}} \newcommand{\thT}{\mathbb{T}} \newcommand{\mod}{\mathbf{Mod}} \newcommand{\th}{\mathbf{Th}} \newcommand{\cl}{\mathbf{Cl}} \newcommand{\ob}{\mathbf{Ob}} \newcommand{\aut}{\mathbf{Aut}} \newcommand{\bun}{\mathbf{Bun}} \newcommand{\geom}{\mathbf{Geom}} \def\catg{\mathop{\mathcal{C}\! {\it at}}\nolimits} \def\Cat{\mathop{\mathfrak{Cat}}} \def\Con{\mathop{\mathfrak{Con}}} \def\CAT{\mathop{\mathbf{2} \mathfrak{Cat}}} \def\Cat{\mathop{\mathfrak{Cat}}} \newcommand{\Topos}{\mathfrak{Top}} \newcommand{\ETopos}{\mathcal{E}\mathfrak{Top}} \newcommand{\BTopos}{\mathcal{B}\mathfrak{Top}} \newcommand{\GTopos}{\mathcal{G}\mathfrak{Top}} \newcommand{\Psh}{\textit{Psh}} \newcommand{\Sh}{\textit{Sh}} \newcommand{\psh}[1]{\textit{Psh}(\cat{#1})} \newcommand{\sh}[1]{\textit{Sh}(\cat{#1})} \newcommand{\Id}{\operatorname{Id}} \newcommand{\Ho}{\operatorname{Ho}} \newcommand{\ad}{\operatorname{ad}} \newcommand{\Adj}{\operatorname{Adj}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Set}{\operatorname{Set}} \newcommand{\Pull}{\operatorname{Pull}} \newcommand{\Push}{\operatorname{Push}} \newcommand{\dom}{\operatorname{dom}} \newcommand{\cod}{\operatorname{cod}} \newcommand{\fun}{\operatorname{Fun}} \newcommand{\colim}{\operatorname{Colim}} \newcommand{\cat}[1]{\mathcal{#1}} \newcommand{\scr}[1]{\mathscr{#1}} \newcommand{\frk}[1]{\mathfrak{#1}} \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\CS}{\mathcal{S}} \newcommand{\CE}{\mathcal{E}} \newcommand{\CF}{\mathcal{F}}\]

First, make sure you check out this amusing list of mathematical short stories and things I was surprised to learn are false on Camarena’ web page.

Jokes

  • A monad over a category $\cat{C}$ is a monoid in the category of endofunctors $[\cat{C}, \cat{C}]$.

  • A monoid is a category with one object and a group is a groupoid with one object.

  • Shall we call categories monoidoids?

  • A category is an interanl monad in the bicategory $Span(\Set)$.

  • A ring is an $Ab$-enriched category with one object.

  • A (right) module $M$ over a ring $R$ is an $Ab$-enriched functor $M\maps *_R \to Ab$.

Factoids

  • The Grothendieck construction of representable presheaves and copresheaves yield slice and coslice categories.

  • For a group (groupoid) $G$, the Grothendieck construction of a $G$-set $X$, considered as a functor $X\maps BG \to \Set$, yields the action group (groupoid).

  • The forgetful functor $\Set_* \to \Set$ has a special status amongs the forgetful functors in the 2-category of categories: it is the universal discrete opfibration. (If you classify laxly then $1 \to \Set$ is the classifier.)

  • The longest series of adjoint functors has six components; one of them is the Yoneda embedding. (Guess which one?)

  • A category is equivalent to a small category if and only if both it and its presheaf category are locally small. (On the Size of Categories, Peter Freyd and Ross Street, TAC 1995)

  • A small category with all (small) limits is equivalent to a complete (and hence cocomplete) lattice.

    • (improvement by J. ADAMEK. H. HERRLICH, AND J. REITERMAN) Any cocomplete weakly co-wellpowered category with a weakly colimit-dense small subcategory is complete.

  • Any slice of a presheaf topos is a presheaf topos and any slice of a sheaf topos is a sheaf topos.

Mathematical terms beginning with “s”:

While attending the school and conference “toposes in Como”, we had several hikes, usually late at night, with Ingo, Mathias, Stuart, Alexander, and few others. In one of these nights, while coming back from our hike, at 3:00 AM, we started a little joke of recalling technical mathematical terms based on their first letter. We noticed the terms beginning with “s” occuring disproportionally more than the others. Ingo did a great effort of listing them in an email a month later. In his email to the group he wrote

below is a list of technical terms in category theory and adjacent subjects which begin with the letter “s”. Extracted from the contents of the nLab, roughly in descending order of number of occurrences on the nLab, cut at some arbitrary point. Some entries are debatable. I omitted the deluge of terms of the form “super” + X, where X is a random word. :-)

space structure set smooth simplicial section sequence string symmetry spectrum standard small super statement spacetime sheaf subset strict system site symplectic subcategory sum square supergravity stack simple series scheme state subspace satisfaction semantics spin strong short singular subgroup synthetic smash sphere shape slice surjective suspension subobject source split support sort simplex selbst science surface star scalar signature separated sieve symbol strength syntax semisimple stability stage subtopos size sober sequent shift substitution scaling self-adjoint substance sheafification story set-theoretic spatial stalk successor sharp supremum sentence second-countable stratification sublocale skeleton subcanonical spacelike sink semifree sifted saturation semi-abelian specialization syllogism submodule second-order semi-locally superposition subgraph simply-connected subquotient semicartesian scope semiring setoid shuffle subnet seminorm syntopogenous S-matrix subsite skew sublated subsingleton semidefinite suplattice subbundle subclass separator snake salamander subformula submersion solid slope subfinite sketch segment subfunctor subbase Stone-Cech Stone space slash syntomic semilattice Scott-continuous sesquicategory submonoid subgroupoid stabilizer subcollection soberification string diagram spread subcoalgebra sesquiunital species