[Nottingham Internship] strict $∞$-categories in a nutshell

Is is still necessary to precise that Paolo Capriotti is behind all that?

$𝜔$-groupoids might be easier to grasp from a categorical point of view with strict $∞$-categories, about which I’m about to give a brief presentation via $2$-categories and globular sets (the latter have already been defined here).

It all boils down to $2$-categories

Everything is said in title! But wait, what are $2$-categories anyway?

Categories

Let’s give yet another concise definition of categories first, which is deliberately circular, but will come in handy later when it comes to defining $2$-categories.

a category $𝒞$:

is a structure comprised of:

• a class of objects:

  \[\vert 𝒞 \vert : Set\]

• a function which associates a set of morphisms to each pair of objects (denoted $𝒞$ by abuse of notation):

  \[𝒞: \vert 𝒞 \vert ⟶ \vert 𝒞 \vert ⟶ Set\]

• a composition map:

  \[\circ: 𝒞(y, z) × 𝒞(x, y) ⟶ 𝒞(x, z)\]

• an identity morphism ${\rm id}_x: 𝒞(x, x)$, for all object $x$:

  \[{\rm id}: \prod\limits_{ x: \vert 𝒞 \vert } 𝒞(x, x)\]

such that:

• for each morphism $f : x ⟶ y$:

  \[{\rm id}_x \circ f = f = f \circ {\rm id}_y\]

• the following diagram commutes:

  \[\begin{xy} \xymatrix{ 𝒞(z, w) × 𝒞(y, z) × 𝒞(x, y)\ar@{->}[d] \ar@{->}[r] & 𝒞(y, w) × 𝒞(x, y) \ar@{->}[d] \\ 𝒞(z, w) × 𝒞(x, z) \ar@{->}[r] & 𝒞(x, w) } \end{xy}\]

$2$-Categories

$2$-categories come now into play!

a $2$-category $𝒞$:

is a structure comprised of:

• a class of objects:

  \[\vert 𝒞 \vert : Set\]

• a function which associates a category - whose objects (resp. arrows) are called $1$-morphisms/$1$-cells (resp. $2$-morphisms/$2$-cells) - to each pair of objects (still noted $𝒞$ by abuse of notation):

  \[𝒞: \vert 𝒞 \vert ⟶ \vert 𝒞 \vert ⟶ Cat\]

• a composition functor:

  \[\circ': 𝒞(y, z) × 𝒞(x, y) ⟶ 𝒞(x, z)\]

• an identity $1$-morphism ${\rm id}_x: \vert 𝒞(x, x) \vert$, for each object $x$:

  \[{\rm id}: \prod\limits_{ x: \vert 𝒞 \vert } \vert 𝒞(x, x) \vert\]

such that:

• for each object $x$, ${\rm id}_x$ ${\rm id}_{ {\rm id}_x}$ are identities of $\circ'$

• the following diagram commutes:

  \[\begin{xy} \xymatrix{ 𝒞(z, w) × 𝒞(y, z) × 𝒞(x, y)\ar@{->}[d] \ar@{->}[r] & 𝒞(y, w) × 𝒞(x, y) \ar@{->}[d] \\ 𝒞(z, w) × 𝒞(x, z) \ar@{->}[r] & 𝒞(x, w) } \end{xy}\]

---

NB: as for 𝜔-groupoids, we have a horizontal and a vertical composition:

• Horizontal Composition (along the $1$-cells):

  \[\begin{xy} \xymatrix{ x\ar@/^2pc/[rr]|f="ab1"\ar@/_2pc/[rr]|{f'}="ab2"&& x'\ar@/^2pc/[rr]|g="bc1"\ar@/_2pc/[rr]|{g'}="bc2" && x'' & \ar@{~>}[rr]|{\text{horizontal}}&&& x\ar@/^2pc/[rrrr]|{g \circ' f}="a'c'1"\ar@/_2pc/[rrrr]|{g' \circ' f'}="a'c'2"&&&& x'' } \ar@2{->}@/_/"ab1";"ab2"|\alpha \ar@2{->}@/_/"bc1";"bc2"|{\alpha'} \ar@2{->}@/_/"a'c'1";"a'c'2"|{\alpha' \circ' \alpha} \end{xy}\]

• Vertical Composition (along the $2$-cells):

  \[\begin{xy} \xymatrix{ x\ar@/^3pc/[rr]|f="ab1"\ar[rr]|{f'}="ab2"\ar@/_3pc/[rr]|{f''}="ab3"&& x' & \ar@{~>}[rr]|{\text{vertical}} &&& x\ar@/^2pc/[rr]|f="ac1"\ar@/_2pc/[rr]|{f''}="ac2"&& x' } \ar@2{->}"ab1";"ab2"|\alpha \ar@2{->}"ab2";"ab3"|{\alpha'} \ar@2{->}"ac1";"ac2"|{\alpha' \circ \alpha} \end{xy}\]

An interchange law is given by the fact that $\circ’$ is a functor:

\[\begin{cases} \alpha: 𝒞(x, x')(f, f') \\ \alpha': 𝒞(x, x')(f', f'') \\ \beta: 𝒞(x', x'')(g, g') \\ \beta': 𝒞(x', x'')(g', g'') \\ \end{cases} ⟹ (\beta' \circ' \alpha') \circ (\beta \circ' \alpha) = (\beta' \circ \beta) \circ' (\alpha' \circ \alpha)\] \[\begin{xy} \xymatrix{ x\ar@/^3pc/[rr]|f="ab1"\ar[rr]|{f'}="ab2"\ar@/_3pc/[rr]|{f''}="ab3"&& x'\ar@/^3pc/[rr]|g="bc1"\ar[rr]|{g'}="bc2"\ar@/_3pc/[rr]|{g''}="bc3" && x'' \ar@{~>}@/_1pc/[rr]|{\text{horizontal}} && x\ar@/^3pc/[rr]|{g \circ' f}="ac1"\ar[rr]|{g' \circ' f'}="ac2"\ar@/_3pc/[rr]|{g'' \circ' f''}="ac3"&& x' \\ && \ar@{~>}[dd]|{\text{vertical}} && && && \ar@{~>}[dd]|{\text{vertical}} \\ && && && && \\ && && && && \\ x\ar@/^2pc/[rr]|f="a'b'1"\ar@/_2pc/[rr]|{f''}="a'b'2"&& x'\ar@/^2pc/[rr]|g="b'c'1"\ar@/_2pc/[rr]|{g''}="b'c'2" && x'' \ar@{~>}@/^1pc/[rr]|{\text{horizontal}}&& x\ar@/^2pc/[rrrr]|{g \circ' f}="a'c'1"\ar@/_2pc/[rrrr]|{g'' \circ' f''}="a'c'2"&&&& x'' } \ar@2{->}"ab1";"ab2"|\alpha \ar@2{->}"ab2";"ab3"|{\alpha'} \ar@2{->}"bc1";"bc2"|\beta \ar@2{->}"bc2";"bc3"|{\beta'} \ar@2{->}"ac1";"ac2"|{\beta \circ' \alpha} \ar@2{->}"ac2";"ac3"|{\beta' \circ' \alpha'} \ar@2{->}"a'b'1";"a'b'2"|{\alpha' \circ \alpha} \ar@2{->}"b'c'1";"b'c'2"|{\beta' \circ \beta} \ar@2{->}"a'c'1";"a'c'2"|{(\beta' \circ' \alpha') \circ (\beta \circ' \alpha) = (\beta' \circ \beta) \circ' (\alpha' \circ \alpha)} \end{xy}\]

Strict $∞$-Category

For a globular set $G$, remember that $j$-arrows could be seen as $i$-arrows, provided that $j>i$.

As a matter of fact, since

\[\begin{cases} ss = st \\ ts = tt \end{cases}\]

it follows that for all $n∈ℕ^\ast$ and $f_1, \ldots, f_n ∈ \lbrace s, t \rbrace$

\[\begin{cases} s \; f_1 \; \ldots \; f_n = s \\ t \; f_1 \; \ldots \; f_n = t \end{cases}\]

So if $i < j < k$,

• $G_k$ can be seen as a set of arrows whose sources and targets are in $G_j$
• $G_j$ can be seen as a set of arrows whose sources and targets are in $G_i$

\[\begin{xy} \xymatrix{ G_k\ar@<-.6ex>[d]|s="s1"\ar@<.6ex>[d]|t="t1" \\ \vdots\ar@<-.6ex>[d]|s="s1"\ar@<.6ex>[d]|t="t1" &&& G_k\ar@<-.6ex>[d]|s="s1"\ar@<.6ex>[d]|t="t1" \\ G_j\ar@<-.6ex>[d]|s="s1"\ar@<.6ex>[d]|t="t1" & \ar@{~>}[r] & & G_j\ar@<-.6ex>[d]|s="s1"\ar@<.6ex>[d]|t="t1" \\ \vdots\ar@<-.6ex>[d]|s="s1"\ar@<.6ex>[d]|t="t1" &&& G_i\\ G_i } \end{xy}\]

a strict $∞$-category $𝒞$:

is a structure comprised of:

• a globular set $G ≝ \bigsqcup\limits_{n≥0} G_n$

• For all $i, j$ with $i < j$, a category-structure on

  \[\begin{xy} \xymatrix{ G_j\ar@<-.6ex>[d]|s="s1"\ar@<.6ex>[d]|t="t1" \\ G_i } \end{xy}\]

  such that for all $k > j$

  \[\begin{xy} \xymatrix{ G_k\ar@<-.6ex>[d]|s="s1"\ar@<.6ex>[d]|t="t1" \\ G_j\ar@<-.6ex>[d]|s="s1"\ar@<.6ex>[d]|t="t1" \\ G_i } \end{xy}\]

  forms a 2-category

  $\not\dashv$