--- date: 2020-11-28 18:45 modification date: Friday 25th March 2022 19:54:33 title: "Introduction to infinity categories" aliases: ["Introduction to infinity categories"] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags - #web/blog #projects/notes/reading #higher-algebra/infty-cats - Refs: - [https://www.youtube.com/watch?v=3IjAy0gHRyY](https://www.youtube.com/watch?v=3IjAy0gHRyY) - Links: - [infinity category](Unsorted/infinity%20categories.md) --- # Introduction to [infinity categories](infinity%20categories.md) > These are notes roughly transcribed from [https://www.youtube.com/watch?v=3IjAy0gHRyY](https://www.youtube.com/watch?v=3IjAy0gHRyY) ## Preliminary Definitions Dealing with size issues: take a [Grothendieck universe](Grothendieck%20universe) $\mathcal{U}$, which are the sets whose subsets are closed under all of the usual set operations. :::{.remark} [Kan complexes](Kan%20complexes) are a specialized notion of [infinity categories](Unsorted/infinity%20categories.md). ::: :::{.definition title="Functors between $\infty\dash$categories"} A **functor** between two $\infty\dash$categories is a morphism of [simplicial sets](simplicial%20sets). ::: :::{.remark} For $\mathcal{C}$ an $\infty\dash$category, we can define $\mathcal{C}_0$ to be the "objects" and $\mathcal{C}_1$ to be the "morphisms", although we don't have a good notion of composition yet. There will be boundary map: a 1-simplex has two boundary points, i.e. two objects $a, b \in \mathcal{C}_0$, so we can think of this as a map $f: a\to b$ where $a = \del_1 f, b= \del_0 f$[^on_bd_notation] are the first and second vertices respectively. We'll also have "degeneracy" maps going up from $\mathcal{C}_0 \to \mathcal{C}_1$, which we should think of as assigning identity morphisms to objects, or conversely that the identity morphism is the degenerate 1-simplex at an object. [^on_bd_notation]: This notation $\bd_i$ denotes the boundary operator that drops the $i$th vertex. ::: ## Equivalences :::{.definition title="Equivalence of Morphisms"} Given two morphisms $f, g: a\to b$ in an $\infty\dash$category, we say $f\homotopic g$ are **equivalent** iff there is a 2-simplex filling in the following diagram: \begin{tikzcd} && b \\ \\ a && b \arrow["{\id_b}", from=1-3, to=3-3] \arrow[""{name=0, anchor=center, inner sep=0}, "f", from=3-1, to=1-3] \arrow["g"', from=3-1, to=3-3] \arrow[shorten <=4pt, shorten >=4pt, Rightarrow, from=0, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJhIl0sWzIsMCwiYiJdLFsyLDIsImIiXSxbMSwyLCJcXGlkX2IiXSxbMCwxLCJmIl0sWzAsMiwiZyIsMl0sWzQsMiwiIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjE1LCJ0YXJnZXQiOjE1fX1dXQ==) ::: :::{.remark} This turns out to be an equivalence relation. Note that in an ordinary category, if two morphisms are equivalent then they are already equal. ::: :::{.definition title="Composition of morphisms"} For 1-simplices $f: a\to b, g:b\to c$, a **composition** of $f$ and $g$ is a 2-simplex $\sigma$ filling in the following diagram: \begin{tikzcd} && {b} \\ \\ {a} &&&& {c} \arrow["{f}", from=3-1, to=1-3] \arrow["{g}", from=1-3, to=3-5] \arrow["{\exists h}"' {name=0, inner sep=0}, from=3-1, to=3-5, dotted] \arrow[Rightarrow, from=0, to=1-3, shorten <=6pt, shorten >=6pt, "\sigma"] \end{tikzcd} > [Link to diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJhIl0sWzQsMiwiYyJdLFsyLDAsImIiXSxbMCwyLCJmIl0sWzIsMSwiZyJdLFswLDFdLFs1LDIsIiIsMCx7Imxlbmd0aCI6NzB9XV0=) In this case, $h \da \bd_1 \sigma$ and we write $h \homotopic g\circ f$. ::: :::{.remark} Note that we're not fixing a choice, but it is well-defined up to the equivalence relation we're using. This is similar to how e.g. coproducts are not baked into the structure of a category, but are instead only well-defined up to canonical isomorphism -- and in fact, this characterization is sometimes preferable. ::: :::{.definition title="Equivalences of objects"} If $f: a\to b$ is a morphism in an $\infty\dash$category $\mathcal{C}$, then we say $f$ is an **equivalence** if there exists a morphism $g:b\to a$ such that $\id_a \homotopic g\circ f$ and $\id_b \homotopic f\circ g$. This is equivalent to finding 2-simplices $\sigma, \sigma'$ that fill the following two diagrams: \begin{tikzcd} && {b} \\ &&&&& {} \\ {a} && {a} && {b} \arrow["{f}"{name=0}, from=3-1, to=1-3] \arrow["{\exists g}", from=1-3, to=3-3, dashed] \arrow["{\id_a}"', from=3-1, to=3-3] \arrow["{\id_b}"{name=1}, from=1-3, to=3-5] \arrow["{\exists g}", from=3-5, to=3-3, dashed] \arrow[Rightarrow, "{\sigma}"', from=0, to=3-3, shorten <=4pt, shorten >=4pt] \arrow[Rightarrow, "{\sigma'}", from=1, to=3-3, shorten <=4pt, shorten >=4pt] \end{tikzcd} > [Link to diagram](https://q.uiver.app/?q=WzAsNSxbMCwyLCJhIl0sWzIsMCwiYiJdLFsyLDIsImEiXSxbNSwxXSxbNCwyLCJiIl0sWzAsMSwiZiJdLFsxLDIsIlxcZXhpc3RzIGciLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwyLCJcXGlkX2EiLDJdLFsxLDQsIlxcaWRfYiJdLFs0LDIsIlxcZXhpc3RzIGciLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNSwyLCJcXHNpZ21hIiwyLHsibGVuZ3RoIjo3MH1dLFs4LDIsIlxcc2lnbWEnIiwwLHsibGVuZ3RoIjo3MH1dXQ==) ::: :::{.remark} This is close to what we'd require for an isomorphism in an ordinary category, but we now allow the compositions to only be "weakly equivalent" or homotopic to the identities. ::: :::{.definition title="Functor Categories"} For $\mathcal{C}, \mathcal{D}$ simplicial sets, we can define a simplicial set $\Fun(\mathcal{C}, \mathcal{D})$ whose $n\dash$simplices are given by \[ \Fun(\mathcal{C}, \mathcal{D})_n \da \ts{ \text{Simplicial maps } F: \mathcal{C} \cross \Delta^n \to \mathcal{D}} .\] ::: :::{.remark} Note that the 0-simplices recover functors if these are ordinary categories. If $\mathcal{D}$ is an $\infty\dash$category, then this functor category is again an $\infty\dash$category. ::: :::{.definition title="Morphisms of functors / natural transformations"} A **morphism** in $\Fun(\mathcal{C}, \mathcal{D})$, say $\eta: F\to G$, is a functor $\eta: C\cross \Delta^n \to \mathcal{D}$ such that \[ \ro{\eta}{\mathcal{C} \cross \ts{0}} &= F \\ \ro{\eta}{\mathcal{C} \cross \ts{1}} &= G .\] We call such an $\eta$ a **natural transformation** from $F$ to $G$. ::: :::{.remark} Being an equivalence in $\Fun(\mathcal{C}, \mathcal{D})$ is equivalent to being a pointwise equivalence. I.e., $\eta$ is an equivalence iff the map $\eta_{{C}}$ given by partially applying an object of $\mathcal{C}$ (i.e. a 1-simplex $\Delta^n \to \mathcal{D}$) is an equivalence in $\mathcal{D}$ for all objects $C\in \Ob(\mathcal{C})$. ::: :::{.definition title="Equivalences of $\infty\dash$categories"} A functor $f:\mathcal{C}\to \mathcal{D}$ of $\infty\dash$categories is an **equivalence** iff there exists a functor $g: \mathcal{D}\to \mathcal{C}$ and natural equivalences \[ f\circ g &\mapsvia{\sim} \id_{\mathcal{D}} \\ g\circ f &\mapsvia{\sim} \id_{\mathcal{C}} .\] If there exists such an equivalence, we will write $\mathcal{C}\homotopic \mathcal{D}$. ::: :::{.remark} For ordinary categories, there is a characteristic property that is much easier to write down in general than an explicit equivalence, namely being essentially surjective and fully faithful. We need the notion of [mapping spaces](mapping%20spaces) to make that precise here. ::: ## Composition [mapping spaces](mapping%20spaces) :::{.definition title="Mapping Spaces"} For $a, b\in \Ob(\mathcal{C})$, we define a simplicial set $\Map_{\mathcal{C}}(a, b)$ as the following pullback: \begin{tikzcd} {Map(a, b)} && {\Fun(\Delta^n, \mathcal{C})} \\ & {} \\ {\Delta^0} && {\mathcal{C} \cross \mathcal{C}} \arrow[from=1-3, to=3-3] \arrow["{(a, b)}"', from=3-1, to=3-3] \arrow[dashed, from=1-1, to=3-1] \arrow[dashed, from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJNYXAoYSwgYikiXSxbMiwwLCJcXEZ1bihcXERlbHRhXm4sIFxcbWF0aGNhbHtDfSkiXSxbMCwyLCJcXERlbHRhXjAiXSxbMiwyLCJcXG1hdGhjYWx7Q30gXFxjcm9zcyBcXG1hdGhjYWx7Q30iXSxbMSwxXSxbMSwzXSxbMiwzLCIoYSwgYikiLDJdLFswLDIsIiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDEsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDQsIiIsMCx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ==) ::: :::{.remark} Here we use the fact if $F\in \Fun(\Delta^1, \mathcal{C})$, this data includes two maps $f, g: \Delta^0 \to \mathcal{C}$ given by restricting to the two vertices of $\Delta^1$. This allows us to define a map $(f, g)$ into $\mathcal{C}^2$. In that product, we also have the point $a, b$, which allows defining the bottom map $(a, b)$. Also note that if $\mathcal{C}$ is in fact an $\infty\dash$category, then $\Map_{\mathcal{C}}(a, b)$ is a Kan complex. The 0-simplices in it are precisely the morphisms in $\Fun(\Delta^n, \mathcal{C})$ with endpoints $a, b$, and there is a filling 1-simplex between any two such morphisms iff they are equivalent. We can thus conclude that \[ \pi_0 \Map_{\mathcal{C}}(a, b) = \ts{\text{Equivalence classes of morphism } f:a\to b} .\] ::: :::{.definition title="Fully Faithful"} A functor $f: \mathcal{C} \to \mathcal{D}$ is **fully faithful** the induced maps \[ f_*: \Map_{\mathcal{C}}(a, b) \to \Map_{\mathcal{D}}(f(a), f(b)) \] are homotopy equivalences of Kan complexes for all pairs of objects $a, b\in \Ob(\mathcal{C})$. ::: :::{.remark} Note that this does imply bijections on (equivalence classes) of morphisms in hom sets, i.e. on $\pi_0$, but in general this is much more because we are requiring an induced isomorphism on all higher homotopy groups as well. ::: :::{.warnings} This is not something that can easily be checked on *just* morphisms. ::: :::{.definition title="Essentially Surjective"} A functor $f:\mathcal{C}\to \mathcal{D}$ is **essentially surjective** iff for every $d\in \mathcal{D}$, there exists an object $c\in \mathcal{C}$ and an equivalence $d \homotopic f(c)$. ::: :::{.theorem title="Characterization of equivalence of $\infty\dash$categories"} A functor $f: \mathcal{C}\to \mathcal{D}$ is an equivalence iff $f$ is fully faithful and essentially surjective. ::: :::{.definition title="Full Subcategories"} Let $S \subset \mathcal{C}_0$ be some subset of objects, and define $\mathcal{C}_S \subset \mathcal{C}$ as a simplicial subset given by \[ \mathcal{C}_S \da \ts{\text{All simplices with vertices in } S} .\] ::: :::{.remark} That this is an $\infty\dash$category follows from checking definitions. If we first *saturate* $S$ under equivalence of objects, i.e. form the larger subset $\bar S \supseteq S$ consisting of all objects in $\mathcal{C}$ which are equivalent to some object in $S$, this produces a functor \[ F: \mathcal{C}_S \injects \mathcal{C}_{\bar S} ,\] which is fully faithful and essentially surjective[^why_obvious] and thus an equivalence. So if you're interested in categories up to equivalence, this replacement is always a valid move. [^why_obvious]: This is purportedly "obvious": being essentially surjective is clear, and fully faithful follows from defining mapping spaces as pullbacks, and writing it out yields an equality of simplicial sets. ::: ## Homotopic (and Contractible Spaces of) Choices :::{.remark} Note that the pullback construction from before seems to generalize: \begin{tikzcd} {\Map_{\mathcal{C}}\qty{\ts{a_1, \cdots, a_n}} } && {\Fun(\Delta^{n-1}, \mathcal{C})} \\ & {} \\ {\Delta^0} && {\mathcal{C}^n} \arrow[from=1-3, to=3-3] \arrow[from=3-1, to=3-3] \arrow[from=1-1, to=3-1, dashed] \arrow[from=1-1, to=1-3, dashed] \arrow["\lrcorner"{very near start, rotate=0}, from=1-1, to=2-2, phantom] \end{tikzcd} > [Link to diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJNYXAoYSwgYikiXSxbMiwwLCJcXEZ1bihcXERlbHRhXm4sIFxcbWF0aGNhbHtDfSkiXSxbMCwyLCJcXERlbHRhXjAiXSxbMiwyLCJcXG1hdGhjYWx7Q30gXFxjcm9zcyBcXG1hdGhjYWx7Q30iXSxbMSwxXSxbMSwzXSxbMiwzLCIoYSwgYikiLDJdLFswLDIsIiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDEsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDQsIiIsMCx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ==) This can be thought of as the space of $n\dash$simplices whose vertices are at the $n+1$ given objects. We can define compositions of morphisms by taking $n=3$, and applying boundary operators yields maps \begin{tikzcd} && {\Map_{\mathcal C}(a,b,c)} \\ \\ {\Map_{\mathcal C}(b,c) \cross \Map_{\mathcal C}(a, b)} &&&& {\Map_{\mathcal C}(a, c)} \arrow["{\bd_1}", from=1-3, to=3-5] \arrow["{f = (\bd_0, \bd_2)}"', from=1-3, to=3-1] \arrow["{\exists h}"', from=3-1, to=3-5, dashed] \end{tikzcd} > [Link to diagram](https://q.uiver.app/?q=WzAsMyxbMiwwLCJcXE1hcF97XFxtYXRoY2FsIEN9KGEsYixjKSJdLFs0LDIsIlxcTWFwX3tcXG1hdGhjYWwgQ30oYSwgYykiXSxbMCwyLCJcXE1hcF97XFxtYXRoY2FsIEN9KGIsYykgXFxjcm9zcyBcXE1hcF97XFxtYXRoY2FsIEN9KGEsIGIpIl0sWzAsMSwiXFxiZF8xIl0sWzAsMiwiZiA9IChcXGJkXzAsIFxcYmRfMikiLDJdLFsyLDEsIlxcZXhpc3RzIGgiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=) where the existence of $h$ follows from the fact that $f$ is an equivalence and can thus be inverted. This is induced by maps on Kan complexes \begin{tikzcd} && {\Delta^2} \\ \\ {\Lambda_1^2} &&&& {\Delta^1} \arrow[from=3-1, to=1-3] \arrow[from=3-5, to=1-3, hook'] \arrow[from=3-1, to=3-5, dashed] \end{tikzcd} > [Link to diagram](https://q.uiver.app/?q=WzAsMyxbMiwwLCJcXERlbHRhXjIiXSxbMCwyLCJcXExhbWJkYV8xXjIiXSxbNCwyLCJcXERlbHRhXjEiXSxbMSwwXSxbMiwwLCIiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6ImJvdHRvbSJ9fX1dLFsxLDIsIiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==) where we're taking the inner horn and the outer face respectively. This can be thought of as horn-filling in families. ::: :::{.remark} **Why is this important?** Given two morphisms, we can pick a composition, and there are multiple ways to do so. We can then look at the middle face to define the actual composition, up to equivalence of morphisms. This relies on a choice of homotopy inverse $s$, allowing us to define a map $\circ_s$. But given an equivalence, there is a unique homotopy inverse up to homotopy, so any two choices of $s$, say $s$ and $s'$ give homotopic maps $\circ_s$ and $\circ_{s'}$. In good situations, we have even more: the space of such choices will be [contractible choice](contractible choice.md), which is stronger than there just being a homotopy between any two choices. So composition is "unique", it's just that there's not one preferred choice. ::: :::{.remark} Associativity follows from a similar line of reasoning applied to $\Map_{\mathcal{C}}(a,b,c,d)$ on four objects. Compare this to [Segal categories](Segal categories), where such spaces are part of the data: categories weakly enriched in spaces, and $\infty\dash$categories recover this for free. There is a way to think of $\infty\dash$categories as "categories enriched in Kan complexes" with a more strict condition of associativity. ::: :::{.remark} We recover all of ordinary category theory when the mapping spaces are discrete. Looking at Kan complexes also yields $\infty\dash$categories where all of the morphisms are invertible, so these are in fact [infinity groupoids](infinity groupoids.md). For us, "spaces" and Kan complexes are synonymous, and the $\infty\dash$category of spaces will be the fundamental example we run with. :::