Definition df-termc 49977

Description: Definition of the proper class (termcnex 50080) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 49990). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 50019). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 50024). See also dftermc3 50035 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 50035, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 49995.

Followed directly from the definition, terminal categories are thin (termcthin 49981). The opposite category of a terminal category is "almost" itself (oppctermco 50009). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 50044).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 50005) and terminal objects (dftermo4 50006). The universal properties provide an alternate proof of initoeu1 17973, termoeu1 17980, initoeu2 17978, and termoeu2 49742. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 50020).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17649, 0thincg 49962, func0g 49593, 0funcg 49589, and initc 49595.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 49976	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1547	. . . . . 6 class 𝑐
4		cbs 17174	. . . . . 6 class Base
5	3, 4	cfv 6489	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1547	. . . . . 6 class 𝑥
8	7	csn 4558	. . . . 5 class {𝑥}
9	5, 8	wceq 1548	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1787	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 49921	. . 3 class ThinCat
12	10, 2, 11	crab 3393	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1548	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is referenced by:  istermc  49978